# Thomas Calculus 12

## Educators

Problem 1

Finding Terms of a Sequence
Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$

$$a_{n}=\frac{1-n}{n^{2}}$$

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Problem 2

Finding Terms of a Sequence
Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$
$$a_{n}=\frac{1}{n !}$$

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Problem 3

Finding Terms of a Sequence
Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$
$$a_{n}=\frac{(-1)^{n+1}}{2 n-1}$$

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Problem 4

Finding Terms of a Sequence
Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$
$$a_{n}=2+(-1)^{n}$$

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Problem 5

Finding Terms of a Sequence
Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$
$$a_{n}=\frac{2^{n}}{2^{n+1}}$$

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Problem 6

Finding Terms of a Sequence
Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$
$$a_{n}=\frac{2^{n}-1}{2^{n}}$$

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Problem 7

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$a_{1}=1, \quad a_{n+1}=a_{n}+\left(1 / 2^{n}\right)$$

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Problem 8

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.
$$a_{1}=1, \quad a_{n+1}=a_{n} /(n+1)$$

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Problem 9

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.
$$a_{1}=2, \quad a_{n+1}=(-1)^{n+1} a_{n} / 2$$

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Problem 10

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.
$$a_{1}=-2, \quad a_{n+1}=n a_{n} /(n+1)$$

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Problem 11

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.
$$a_{1}=a_{2}=1, \quad a_{n+2}=a_{n+1}+a_{n}$$

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Problem 12

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.
$$a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{\mathrm{n}}$$

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Problem 13

Finding a Sequence's Formula In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

The sequence $1,-1,1,-1,1, \ldots$

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Problem 14

In Exercises 13-26, find a formula for the nth term of the sequence.
The sequence $-1,1,-1,1,-1, \ldots$

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Problem 15

In Exercises 13-26, find a formula for the nth term of the sequence.
The sequence $$1,-4,9,-16,25, \dots Check back soon! Problem 16 In Exercises 13-26, find a formula for the nth term of the sequence. The sequence$$1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}, \dots$$Check back soon! Problem 17 In Exercises 13-26, find a formula for the n th term of the sequence.$$\frac{1}{9}, \frac{2}{12}, \frac{2^{2}}{15}, \frac{2^{3}}{18}, \frac{2^{4}}{21}, \dots$$Check back soon! Problem 18 In Exercises 13-26, find a formula for the n th term of the sequence.$$-\frac{3}{2},-\frac{1}{6}, \frac{1}{12}, \frac{3}{20}, \frac{5}{30}, \dots$$Check back soon! Problem 19 In Exercises 13-26, find a formula for the n th term of the sequence.$$0,3,8,15,24, \ldots$$Check back soon! Problem 20 In Exercises 13-26, find a formula for the n th term of the sequence. The sequence$$-3,-2,-1,0,1$$Check back soon! Problem 21 In Exercises 13-26, find a formula for the n th term of the sequence. The sequence$$1,5,9,13,17, \ldots$$Check back soon! Problem 22 In Exercises 13-26, find a formula for the n th term of the sequence. The sequence$$2,6,10,14,18, \dots$$Check back soon! Problem 22 In Exercises 13-26, find a formula for the n th term of the sequence. The sequence$$2,6,10,14,18, \dots$$Check back soon! Problem 23 In Exercises 13-26, find a formula for the n th term of the sequence.$$\frac{5}{1}, \frac{8}{2}, \frac{11}{6}, \frac{14}{24}, \frac{17}{120}, \ldots$$Check back soon! Problem 24 In Exercises 13-26, find a formula for the n th term of the sequence.$$\frac{1}{25}, \frac{8}{125}, \frac{27}{625}, \frac{64}{3125}, \frac{125}{15,625}$$Check back soon! Problem 25 In Exercises 13-26, find a formula for the n th term of the sequence.$$1,0,1,0,1, \ldots$$Check back soon! Problem 26 In Exercises 13-26, find a formula for the n th term of the sequence.$$0,1,1,2,2,3,3,4, \ldots$$Check back soon! Problem 27 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=2+(0.1)^{n}$$Check back soon! Problem 28 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n+(-1)^{n}}{n}$$Check back soon! Problem 29 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{1-2 n}{1+2 n}$$Check back soon! Problem 30 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{2 n+1}{1-3 \sqrt{n}}$$Check back soon! Problem 31 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$\alpha_{n}=\frac{1-5 n^{4}}{n^{4}+8 n^{3}}$$Check back soon! Problem 32 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n+3}{n^{2}+5 n+6}$$Check back soon! Problem 33 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n^{2}-2 n+1}{n-1}$$Check back soon! Problem 34 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{1-n^{3}}{70-4 n^{2}}$$Check back soon! Problem 35 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=1+(-1)^{n}$$Check back soon! Problem 36 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=(-1)^{n}\left(1-\frac{1}{n}\right)$$Check back soon! Problem 37 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{n+1}{2 n}\right)\left(1-\frac{1}{n}\right)$$Check back soon! Problem 38 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(2-\frac{1}{2^{n}}\right)\left(3+\frac{1}{2^{n}}\right)$$Check back soon! Problem 39 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{(-1)^{n+1}}{2 n-1}$$Check back soon! Problem 40 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(-\frac{1}{2}\right)^{n}$$Check back soon! Problem 41 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt{\frac{2 n}{n+1}}$$Check back soon! Problem 42 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{1}{(0.9)^{n}}$$Check back soon! Problem 43 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sin \left(\frac{\pi}{2}+\frac{1}{n}\right)$$Check back soon! Problem 44 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=n \pi \cos (n \pi)$$Check back soon! Problem 45 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{\sin n}{n}$$Check back soon! Problem 46 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{\sin ^{2} n}{2^{n}}$$Check back soon! Problem 47 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n}{2^{n}}$$Check back soon! Problem 48 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{3^{n}}{n^{3}}$$Check back soon! Problem 49 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{\ln (n+1)}{\sqrt{n}}$$Check back soon! Problem 50 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{\ln n}{\ln 2 n}$$Check back soon! Problem 51 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=8^{1 / n}$$Check back soon! Problem 52 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=(0.03)^{1 / n}$$Check back soon! Problem 53 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(1+\frac{7}{n}\right)^{n}$$Check back soon! Problem 54 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(1-\frac{1}{n}\right)^{n}$$Check back soon! Problem 55 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt[n]{10 n}$$Check back soon! Problem 56 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt[n]{n^{2}}$$Check back soon! Problem 57 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{3}{n}\right)^{1 / n}$$Check back soon! Problem 58 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=(n+4)^{1 /(n+4)}$$Check back soon! Problem 59 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{\ln n}{n^{1 / n}}$$Check back soon! Problem 60 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\ln n-\ln (n+1)$$Check back soon! Problem 61 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt[n]{4^{n} n}$$Check back soon! Problem 62 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt[n]{3^{2 n+1}}$$Check back soon! Problem 63 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n !}{n^{n}}(\text {Hint} \text { : Compare with } 1 / n .)$$Check back soon! Problem 64 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{(-4)^{n}}{n !}$$Check back soon! Problem 65 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n !}{10^{6 n}}$$Check back soon! Problem 66 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n !}{2^{n} \cdot 3^{n}}$$Check back soon! Problem 67 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{1}{n}\right)^{1 / \ln \pi)}$$Check back soon! Problem 68 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\ln \left(1+\frac{1}{n}\right)^{n}$$Check back soon! Problem 69 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{3 n+1}{3 n-1}\right)^{n}$$Check back soon! Problem 70 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{n}{n+1}\right)^{n}$$Check back soon! Problem 71 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{x^{n}}{2 n+1}\right)^{1 / n}, \quad x>0$$Check back soon! Problem 72 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(1-\frac{1}{n^{2}}\right)^{n}$$Check back soon! Problem 74 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{(10 / 11)^{n}}{(9 / 10)^{n}+(11 / 12)^{n}}$$Check back soon! Problem 75 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\tanh n$$Check back soon! Problem 76 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sinh (\ln n)$$Check back soon! Problem 77 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{n^{2}}{2 n-1} \sin \frac{1}{n}$$Check back soon! Problem 78 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=n\left(1-\cos \frac{1}{n}\right)$$Check back soon! Problem 79 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt{n} \sin \frac{1}{\sqrt{n}}$$Check back soon! Problem 80 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(3^{n}+5^{n}\right)^{1 / n}$$Check back soon! Problem 81 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\tan ^{-1} n$$Check back soon! Problem 82 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{1}{\sqrt{m}} \tan ^{-1} n$$Check back soon! Problem 83 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\left(\frac{1}{3}\right)^{n}+\frac{1}{\sqrt{2^{n}}}$$Check back soon! Problem 84 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\sqrt[n]{n^{2}+n}$$Check back soon! Problem 85 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{(\ln n)^{200}}{n}$$Check back soon! Problem 86 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{(\ln n)^{5}}{\sqrt{n}}$$Check back soon! Problem 87 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=n-\sqrt{n^{2}-n}$$Check back soon! Problem 88 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{1}{\sqrt{n^{2}-1}-\sqrt{n^{2}+n}}$$Check back soon! Problem 89 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\frac{1}{n} \int_{1}^{n} \frac{1}{x} d x$$Check back soon! Problem 90 Which of the sequences \left\{a_{n}\right\} in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence.$$a_{n}=\int_{1}^{n} \frac{1}{x^{p}} d x, \quad p>1$$Check back soon! Problem 91 In Exercises 91-98, assume that each sequence converges and find its limit.$$a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}}$$Check back soon! Problem 92 In Exercises 91-98, assume that each sequence converges and find its limit.$$a_{1}=-1, \quad a_{n+1}=\frac{a_{n}+6}{a_{n}+2}$$Check back soon! Problem 93 In Exercises 91-98, assume that each sequence converges and find its limit.$$a_{1}=-4, \quad a_{n+1}=\sqrt{8+2 a_{n}}$$Check back soon! Problem 94 In Exercises 91-98, assume that each sequence converges and find its limit.$$a_{1}=0, \quad a_{n+1}=\sqrt{8+2 a_{n}}$$Check back soon! Problem 95 In Exercises 91-98, assume that each sequence converges and find its limit.$$a_{1}=5, \quad a_{n+1}=\sqrt{5} a_{n}$$Check back soon! Problem 96 In Exercises 91-98, assume that each sequence converges and find its limit.$$a_{1}=3, \quad a_{n+1}=12-\sqrt{a_{n}}$$Check back soon! Problem 97 In Exercises 91-98, assume that each sequence converges and find its limit.$$2,2+\frac{1}{2}, 2+\frac{1}{2+\frac{1}{2}}, 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, \dots$$Check back soon! Problem 98 In Exercises 91-98, assume that each sequence converges and find its limit.$$\sqrt{1}, \sqrt{1+\sqrt{1}}, \sqrt{1+\sqrt{1+\sqrt{1}}}\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}$$Check back soon! Problem 99 The first term of a sequence is x_{1}=1 . Each succeeding term is the sum of all those that come before it:$$x_{n+1}=x_{1}+x_{2}+\dots+x_{n}$$Write out enough early terms of the sequence to deduce a general formula for x_{n} that holds for n \equiv 2 . Check back soon! Problem 100 A sequence of rational numbers is described as follows:$$\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \ldots, \frac{a}{b}, \frac{a+2 b}{a+b}, \ldots$$Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let x_{n} and y_{n} be, respectively, the numerator and the denominator of the m th fraction r_{n}=x_{n} / y_{n} $$\begin{array}{l}{\text { a. Verify that } x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1 \text { and, more }} \\ {\text { generally, that if } a^{2}-2 b^{2}=-1 \text { or }+1, \text { then }}\end{array}$$$$(a+2 b)^{2}-2(a+b)^{2}=+1 \quad \text { or } \quad-1$$respectively. $$\begin{array}{l}{\text { b. The fractions } r_{n}=x_{n} / y_{n} \text { approach a limit as } n \text { increases. }} \\ {\text { What is that limit? (Hint: Use part (a) to show that }} \\ {r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2} \text { and that } y_{n} \text { is not less than } n \text { . }}\end{array}$$ Check back soon! Problem 101 Newton's method The following sequences come from the recursion formula for Newton's method, x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} Do the sequences converge? If so, to what value? In each case, begin by identifying the function f that generates the sequence. a. x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{\mathrm{m}}} b. x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}} c. x_{0}=1, \quad x_{n+1}=x_{2}-1 Check back soon! Problem 102 $$\begin{array}{l}{\text { a. Suppose that } f(x) \text { is differentiable for all } x \text { in }[0,1] \text { and that }} \\ {f(0)=0 . \text { Define sequence }\left\{a_{n}\right\} \text { by the rule } a_{n}=n f(1 / n) \text { . }} \\ {\text { Show that lim }_{n \rightarrow \infty} a_{n}=f^{\prime}(0) . \text { Use the result in part (a) to }} \\ {\text { find the limits of the following sequences }\left\{a_{n}\right\}}\end{array} \\ {\text { b. } a_{n}=n \tan ^{-1} \frac{1}{n} \quad \text { c. } a_{n}=n\left(e^{1 / n}-1\right)} \\ {\text { d. } a_{n}=n \ln \left(1+\frac{2}{n}\right)}$$ Check back soon! Problem 103 Pythagorean triples A triple of positive integers a, b, and c is called a Pythagorean triple if a^{2}+b^{2}=c^{2} . Let a be an odd positive integer and let b=\left\lfloor\frac{a^{2}}{2}\right\rfloor and  c=\left\lceil\frac{a^{2}}{2}\right\rceil be, respectively, the integer floor and ceiling for a^{2} / 2 a. Show that a^{2}+b^{2}=c^{2} . (Hint: Let a=2 n+1 and express b and c in terms of n . ) b. By direct calculation, or by appealing to the accompanying figure, find$$\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}| \right.}{ | \frac{a^{2}}{2} \rceil}$$Check back soon! Problem 104 The n th root of n ! a. Show that \lim _{n \rightarrow \infty}(2 n \pi)^{1 /(2 n)}=1 and hence, using Stirling's approximation (Chapter 8 , Additional Exercise 32 \mathrm{a} , that \sqrt[n]{n !} \approx \frac{n}{a} for large values of n b. Test the approximation in part (a) for n=40,50,60, \ldots, as far as vour calculator will allow. Check back soon! Problem 105 a. Assuming that \lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0 if c is any positive con- stant, show that$$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{c}}=0$$if c is any positive constant. b. Prove that \lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0 if c is any positive constant. (Hint: If \epsilon=0.001 and c=0.04, how large should N be to ensure that \left|1 / n^{c}-0\right|<\epsilon if n>N ? ) Check back soon! Problem 106 The zipper theorem Prove the "zipper theorem" for sequences: If \left\{a_{n}\right\} and \left\{b_{n}\right\} both converge to L, then the sequence a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{n}, b_{n}, \ldots a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}, \ldots Check back soon! Problem 107 Prove that \lim _{n \rightarrow \infty} \sqrt[n]{n}=1 Check back soon! Problem 108 Prove that \lim _{n \rightarrow \infty} x^{1 / n}=1,(x>0) Check back soon! Problem 109 Prove Theorem 2 Check back soon! Problem 110 Prove Theorem 3 Check back soon! Problem 111 In Exercises 111-114, determine if the sequence is monotonic and if it is bounded.$$a_{n}=\frac{3 n+1}{n+1}$$Check back soon! Problem 112 In Exercises 111-114, determine if the sequence is monotonic and if it is bounded.$$a_{n}=\frac{(2 n+3) !}{(n+1) !}$$Check back soon! Problem 113 In Exercises 111-114, determine if the sequence is monotonic and if it is bounded.$$a_{n}=\frac{2^{n} 3^{n}}{n !}$$Check back soon! Problem 114 In Exercises 111-114, determine if the sequence is monotonic and if it is bounded.$$a_{n}=2-\frac{2}{n}-\frac{1}{2^{n}}$$Check back soon! Problem 115 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=1-\frac{1}{n}$$Check back soon! Problem 116 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=n-\frac{1}{n}$$Check back soon! Problem 117 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=\frac{2^{n}-1}{2^{n}}$$Check back soon! Problem 118 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=\frac{2^{n}-1}{3^{n}}$$Check back soon! Problem 119 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right)$$Check back soon! Problem 120 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers. The first term of a sequence is x_{1}=\cos (1) . The next terms are x_{2}=x_{1} or \cos (2), whichever is larger; and x_{3}=x_{2} or \cos (3) , whichever is larger (farther to the right). In general, x_{n+1}=\max \left\{x_{n}, \cos (n+1)\right\} Check back soon! Problem 121 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=\frac{1+\sqrt{2 n}}{\sqrt{n}}$$Check back soon! Problem 122 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=\frac{n+1}{n}$$Check back soon! Problem 123 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{n}=\frac{4^{n+1}+3^{n}}{4^{n}}$$Check back soon! Problem 124 Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your answers.$$a_{1}=1, \quad a_{n+1}=2 a_{n}-3$$Check back soon! Problem 125 The sequence \{n /(n+1)\} has a least upper bound of 1 show that if M is a number less than 1, then the terms of (n /(n+1)\} eventually exceed M . That is, if M<1 there is aninteger N such that n /(n+1)>M whenever n>N . since n /(n+1)<1 for every n, this proves that 1 is a least upper bound for \{n /(n+1)\} Check back soon! Problem 126 Uniqueness of least upper bounds Show that if M_{1} and M_{2} are least upper bounds for the sequence \left\{a_{n}\right\}, then M_{1}=M_{2} . That is, a sequence cannot have two different least upper bounds, Check back soon! Problem 128 Prove that if \left\{a_{n}\right\} is a convergent sequence, then to every positive number \epsilon there corresponds an integer N such that for all m and n, m>N and \quad n>N \Rightarrow\left|a_{m}-a_{n}\right|<\epsilon Check back soon! Problem 129 Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L_{1} and L_{2} are numbers such that a_{n} \rightarrow L_{1} and a_{n} \rightarrow L_{2, \text { then }} L_{1}=L_{2} . Check back soon! Problem 130 Limits and sub sequences If the terms of one sequence appear in another sequence in their given order, we call the first sequence a sub sequence of the second. Prove that if two sub- sequences of a sequence \left\{a_{n}\right\} have different limits L_{1} \neq L_{2} then \left\{a_{y}\right\} diverges. Check back soon! Problem 131 For a sequence \left\{a_{n}\right\} the terms of even index are denoted by a_{2 k} and the terms of odd index by a_{2 k+1} . Prove that if a_{2 k} \rightarrow L and a_{2 k+1} \rightarrow L, then a_{n} \rightarrow L Check back soon! Problem 132 Prove that a sequence \left\{a_{a}\right\} converges to 0 if and only if the sequence of absolute values \left\{\left|a_{n}\right|\right\} converges to 0 . Check back soon! Problem 133 Sequence. generated by N_on'. method Newton's method, applied to a different table function(x), begins with a starting value of and constructs from it a sequence of numbers {x,} that under favorable circumstances converges to a zero of f. The recursion formula for the sequence is x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} a. Show that the recursion formula for f(x)=x^{2}-a, a>0 can be written as x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2 b. Starting with x_{0}=1 and a=3, calculate successive terms of the sequence until the display begins to repeat. What number is being an proximated? Explain Check back soon! Problem 134 A recursive definition of \pi / 2 If you start with x_{1}=1 and define the subsequent terms of \left\{x_{n}\right\} by the rule x_{n}=x_{n-1}+\cos x_{n-1} you generate a sequence that converges rapidly to \pi / 2, (a) Try it. (b) Use the accompanying figure to explain why the convergence is so rapid. Check back soon! Problem 135 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=\sqrt[n]{n}$$Check back soon! Problem 136 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=\left(1+\frac{0.5}{n}\right)$$Check back soon! Problem 137 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{5^{n}}$$Check back soon! Problem 138 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{1}=1, \quad a_{n+1}=a_{n}+(-2)^{n}$$Check back soon! Problem 139 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=\sin n$$Check back soon! Problem 140 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=n \sin \frac{1}{n}$$Check back soon! Problem 141 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=\frac{\sin n}{n}$$Check back soon! Problem 142 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=\frac{\ln n}{n}$$Check back soon! Problem 143 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=(0.9999)^{n}$$Check back soon! Problem 144 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=(123456)^{1 / n}$$Check back soon! Problem 145 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$a_{n}=\frac{8^{n}}{n !}$$Check back soon! Problem 146 Use a CAS to perform the following steps for the sequences in Exerciscs 135-146 . a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? Does it appear to converge or diverge? If it does converge, what is the limit L ? b. If the sequence converges, find an integer N such that \left|a_{\pi}-L\right| \leq 0.01 for n \geq N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L ?$$\alpha_{n}=\frac{n^{41}}{19^{n}}

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