Calculus: Early Transcendentals

(a) What is a sequence?
(b) What does it mean to say that $ \lim_{n \to \infty} a_n = 8? $
(c) What does it mean to say that $ \lim_{n \to \infty} a_n = \infty? $

(a) What is a convergent sequence? Give two examples.
(b) What is a divergent sequence? Give two examples.

List the first five terms of the sequence.
$ a_n = \frac {2^n}{2n + 1} $

.List the first five terms of the sequence.
$ a_n = \frac {n^2 - 1}{n^2 + 1} $

List the first five terms of the sequence.
$ a_n = \frac {(-1)^{n-1}}{5^n} $

List the first five terms of the sequence.
$$ a_n = \cos {n \pi}{2} $$

List the first five terms of the sequence.
$ a_n = \frac {1}{n + 1}! $

List the first five terms of the sequence.
$ a_n = \frac {(-1)^nn}{n! + 1} $

List the first five terms of the sequence.
$ a_1 = 1, a_{n+1} = 5a_n - 3 $

List the first five terms of the sequence.
$ a_1 = 6, a_{n+1} = \frac {a_n}{n} $

List the first five terms of the sequence.
$ a_1 = 2, a_{n+1} = \frac {a_n}{1 + a_n} $

List the first five terms of the sequence.
$ a_1 = 2, a_2 = 1, a_{a+1} = a_n - a_{n-1} $

Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.
$ \left\{\begin{array} \frac {1}{2}, \frac {1}{4}, \frac {1}{6}, \frac {1}{8}, \frac {1}{10}, . . . .\end{array}\right\} $

Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.
$$ \left\{\begin{array} 4, -1, \frac {1}{4}, - \frac {1}{16}, \frac {1}{64}, . . . . .\end{array}\right\} $$

Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.
$ \left\{ -3, 2, - \frac {4}{3}, {8}{9}, - \frac {16}{27}, . . .\right\} $

Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.
$ \left\{\begin{array} 5, 8, 11, 14, 17, . . . . .\end{array}\right\} $

Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.
$ \left\{\begin{array} \frac {1}{2}, - \frac {4}{3}, \frac {9}{4}, - \frac {16}{5}, \frac {25}{6}, . . . . .\end{array}\right\} $

Find a formula for the general term $ a_n $ of the sequence, assuming that the pattern of the first few terms continues.
$ \left\{\begin{array} 1, 0, -1, 0, 1, 0, -1, 0, . . . .\end{array}\right\} $

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$ a_n = \frac {3n}{1 + 6n} $

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$ a_n = 2 + \frac {(-1)^n}{n} $

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$ a_n = 1 + (- \frac {1}{2})^n $

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$ a_n = 1 + \frac{10^n}{9^n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {3 + 5n^2}{n + n^2} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {3 + 5n^2}{1 + n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {n^4}{n^3 - 2n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = 2 + (0.86)^n $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = 3^n 7^{-n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {3 \sqrt {n}}{\sqrt {n} + 2} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = e^{-1/ \sqrt n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \sqrt { \frac {1 + 4n^2}{1 + n^2}} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {4^n}{1 + 9^n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \cos \left( \frac {n \pi}{n + 1} \right) $

Determine whether the sequence converges or diverges. If it converges, find the limit.

$ a_n = \frac {n^2}{\sqrt {n^3 + 4n}} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = e^{2n/(n + 2)} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {(-1)^n}{2 \sqrt n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {(-1)^{n + 1}n}{n + \sqrt n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \left \{ \frac {(2n - 1)!}{(2n + 1)!}\right \}$

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \left \{ \frac {\ln n}{\ln 2n} \right \} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \{ \sin n \} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {\tan^{-1}n}{n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \{ n^2e^{-n}\} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \ln (n + 1) - \ln n $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac { \cos^2 n}{2^n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \sqrt [n]{2^{1 + 3n}} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = n \sin (1/n) $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = 2^{-n} \cos n \pi $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \left( 1+ \frac {2}{n} \right)^n $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \sqrt[n]{n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \ln(2n^2 + 1) - \ln(n^2 + 1) $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac { (\ln n)^2}{n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \arctan (\ln n) $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = n - \sqrt {n + 1} \sqrt {n + 3} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \left \{ 0, 1, 0, 0, 1, 0, 0, 0, 1, . . . \right \} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \left \{ \frac {1}{1}, \frac {1}{3}, \frac {1}{2}, \frac {1}{4}, \frac {1}{3}, \frac {1}{5}, \frac {1}{4}, \frac {1}{6}, . . . \right \} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {n!}{2^n} $

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {(-3)^n}{n!} $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = (-1)^n \frac {n}{n + 1} $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \frac { \sin n}{n} $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \arctan \left( \frac {n^2}{n^2 + 4} \right) $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \sqrt[n]{3^n + 5^n} $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \frac {n^2 \cos n}{1 + n^2} $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{n!} $

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \frac {1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{(2n)^n} $

(a) Determine whether the sequence defined as follows is convergent or divergent:
$ a_1 = 1 $ $ a_{n + 1} = 4 - a_n $ for $ n \ge 1 $

(b) What happens if the first term is $ a_1 = 2 $ ?

If $ \$ $1000 is invested at $ 6 \% $ interest, compounded annually, then after $ n $ years the investment is worth $ a_n = 1000(1.06)^n $ dollars.
(a) Find the first five terms of the sequence $ \{ a_n\}. $
(b) Is the sequence convergent or divergent? Explain.

If you deposit $ \$ $100 at the end of every month into an account that pays $ 3 \% $ interest per year compounded monthly, the amount of interest accumulated after $ n $ months is given by the sequence
$ I_n = 100 \left( \frac {1.0025^n - 1}{0.0025} - n\right) $
(a) Find the first six terms of the sequence.
(b) How much interest will you have earned after two years?

A fish has 5000 catfish in his pond. The number of catfish by $ 8 \% $ per month and the farmer harvests 300 catfish per month.
(a) Show that the catfish population $ P_n $ after $ n $ months is given recursively by
$ P_n = 1.08 P_{n-1} - 300$
$P_0 = 5000$
(b) How many catfish are in the pond after six months?

Find the first 40 terms of the sequence defined by
$ a_{n + 1} =\left\{
\begin{array}{ll}
\frac{1}{2} a_n & \text{if } a_n \text{ is an even number} \\
3a_n + 1 & \text{if } a_n \text{ is an odd number } \end{array} \right. $
and $ a_1 = 11. $ Do the same if $ a_1 = 25. $ Make a conjecture about this type of sequence.

For what values of $ r $ is the sequence $ \left\{ nr^n \right\} $ convergent?

(a) If $ \left \{ a_n \right\} $ is convergent, show that
$ \displaystyle\lim_{n\to\infty} a_{n+1} = \displaystyle\lim_{n\to\infty} a_n $
(b) A sequence $ \left\{ a_n \right\} $ is defined by $ a_1 = 1 $ and $ a_{n + 1} = 1/(1 + a_n) $ for $ n \ge 1. $ Assuming that $ \left\{ a_n \right\} $ is convergent, find its limit.

Suppose you know that $ \left\{ a_n \right\} $ is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = \cos n $

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = \frac{1}{2n + 3} $

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = \frac{1 - n}{2 +n} $

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = n(-1)^n $

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = 2 + \frac{(-1)^n}{n} $

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = 3 - 2ne^{-n} $

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = n^3 - 3n + 3 $

Find the limit of the sequence
$ \left\{ \sqrt 2, \sqrt{2\sqrt2}, \sqrt{2\sqrt{2\sqrt2}}, \cdot \cdot \cdot \right\} $

A sequence $ \left\{ a_n \right\} $ is given by $ a_1 = \sqrt 2, a_{n + 1} = \sqrt {2 + a_n}. $
(a) By induction or otherwise, show that $ \left\{ a_n \right\} $ is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that $ \lim_{n\to\infty} a_n $ exists.
(b) Find $ \lim_{n\to\infty} a_n. $

Show that the sequence defined by
$ a_1 = 1 $
$ a_{n + 1} = 3 - \frac{1}{a_n} $
is increasing and $ a_n < 3 $ for all $ n. $ Deduce that $ \{ a_n \} $ is convergent and find its limit.

Show that the sequence defined by
$ a_1 = 2 $
$ a_{n + 1} = \frac {1}{3 - a_n} $
satisfies $ 0 < a_n \le 2 $ and is decreasing. Deduce that the sequence is convergent and find its limit.

(a) Fibonacci posed the following : Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs or rabbits will we have in the $ n $th month? Show that the answer is $ f_n $ where $ \{ f_n \} $ is the Fibonacci sequence defined in Example 3(c).
(b) Let $ a_n = f_{n + 1} / f_n $ and show that $ a_{n - 1} = 1 + 1/a_{n - 2}. $ Assuming that $ \{ a_n \} $ is convergent, find its limit.

(a) Let $ a_1 =a, a_2 = f(a), a_3 = f(a_2) = f( f(a)), . . . , a_{n + 1} = f(a_n), $ where $ f $ is a continuous function. If $ lim_{n \to\infty} a_n = L, $ show that $ f(L) = L. $
(b) Illustrate part (a) by taking $ f(x) = \cos x, a = 1, $ and estimating the value of $ L $ to five decimal places.

(a) Use a graph to guess the value of the limit
$ \displaystyle \lim_{n \to \infty} \frac {n^5}{n!} $
(b) Use a graph of the sequence in part (a) to find the smallest values of $ N $ that correspond to $ \varepsilon = 0.1 $ and $ \varepsilon = 0.001 $ in Definition 2.

Use Definition 2 directly to prove that $ lim_{n \to \infty} r^n = 0 $ when $ \mid r \mid < 1. $

Prove Theorem 6.
[Hint: Use either Definition 2 or the Squeeze Theorem. ]

Prove Theorem 7.

Prove that if $ \lim_{n \to \infty} a_n = 0 $ and $ \left \{ b_n \right \} $ is bounded, then $ \lim_{n \to\infty} (a_n b_n) = 0. $

Let $ a_n = \left ( 1 + \frac {1}{n} \right)^n. $
(a) Show that if $ 0 \le a < b, $ then
$ \frac {b^{n + 1} - a^{n + 1}}{b -a } < (n + 1)b^n $
(b) Deduce that $ b^n[(n + 1)a - nb] < a^{n + 1}. $
(c) Use $ a = 1 + 1/(n + 1) $ and $ b = 1 + 1/n $ in part (b) to show that $ \left \{ a_n \right \} $ is increasing.
(d) Use $ a = 1 $ and $ b = 1 + 1/(2n) $ in part (b) to show that $ a_{2n} < 4. $
(e) Use parts (c) and (d) to show that $ a_n < 4 $ for all $ n. $
(f) Use Theorem 12 to show that $ \lim_{n \to\infty} (1 + 1/n)^n $ exists.
(The limit is $ e. $ See Equation 3.6.6.)

Let $ a $ and $ b $ be positive numbers with $ a > b. $ Let $ a_1 $ be their arithmetic mean and $ b_1 $ their geometric mean:
$ a_1 = \frac {a + b}{2} $
$ b_1 = \sqrt {ab} $
Repeat this process so that, in general,
$ a^{n + 1} = \frac {a_n + b_n}{2} $
$ b_{n + 1} = \sqrt {a_n b_n} $
(a) Use mathematical induction to show that
$ a_n > a_{n + 1} > b_{n + 1} > b_n $
(b) Deduce that both $ \{ a_n \} $ and $ \{ b_n \} $ are convergent.
(c) Show that $ \lim_{n \to\infty} a_n = \lim_{n \to \infty} b_n $. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers $ a $ and $ b. $

(a) Show that if $ \lim_{n \to \infty} a_{2n} = L $ and $ \lim_{n \to\infty} a_{2n + 1} = L, $ then $ \{ a_n \} $ is convergent and $ \lim_{n \to \infty} a_n = L $.
(b) If $ a_1 = 1 $ and
$ a_{n + 1} = 1 + \frac {1}{1 + a_n} $
find the first eight terms of the sequence $ \{ a_n \} $. Then use part (a) to show that $ \lim_{n \to \infty} a_n = \sqrt{ 2 } $. This gives the continued fraction expansion
$ \sqrt{ 2 } = 1 + \frac {1}{ 2 + \frac {1}{2 + \cdot \cdot \cdot}} $

The size of an undisturbed fish population has been modeled by the formula
$ p_{n + 1} = \frac {bp_n}{a + p_n} $
where $ p_n $ is the fish population after $ n $ years and $ a $ and $ b $ are positive constants that depend on the species and its environment. Suppose that the population in year 0 is $ p_0 > 0. $
(a) Show that if $ \{ p_n \} $ is convergent, then the only possible values for its limit are 0 and $ b - a $.
(b) Show that $ p_{n + 1} < (b/a)p_n $.
(c) Use part (b) to show that if $ a > b, $ then $ \lim_{n \to \infty} p_n = 0 $; in other words, the population dies out.
(d) Now assume that $ a < b $. Show that if $ p_0 < b - a $, then $ \{ p_n \} $ is increasing and $ 0 < p_n < b - a $. Show also that if $ p_0 > b - a $, then $ \{ p_n \} $ is decreasing and $ p_n > b - a $. Deduce that if $ a < b $, then $ \lim_{n \to \infty} p_n = b - a $.