Ask your homework questions to teachers and professors, meet other students, and be entered to win $600 or an Xbox Series X 🎉Join our Discord! ## Chapter 11 ## Infinite Sequences and Series ## Educators GR JS ### Problem 1 (a) What is a sequence? (b) What does it mean to say that$ \lim_{n \to \infty} a_n = 8? $(c) What does it mean to say that$ \lim_{n \to \infty} a_n = \infty? $GR Gabriel R. Numerade Educator ### Problem 2 (a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples. GR Gabriel R. Numerade Educator ### Problem 3 List the first five terms of the sequence.$ a_n = \frac {2^n}{2n + 1} $GR Gabriel R. Numerade Educator ### Problem 4 .List the first five terms of the sequence.$ a_n = \frac {n^2 - 1}{n^2 + 1} $GR Gabriel R. Numerade Educator ### Problem 5 List the first five terms of the sequence.$ a_n = \frac {(-1)^{n-1}}{5^n} $GR Gabriel R. Numerade Educator ### Problem 6 List the first five terms of the sequence. $$a_n = \cos {n \pi}{2}$$ GR Gabriel R. Numerade Educator ### Problem 7 List the first five terms of the sequence.$ a_n = \frac {1}{n + 1}! $GR Gabriel R. Numerade Educator ### Problem 8 List the first five terms of the sequence.$ a_n = \frac {(-1)^nn}{n! + 1} $GR Gabriel R. Numerade Educator ### Problem 9 List the first five terms of the sequence.$ a_1 = 1, a_{n+1} = 5a_n - 3 $GR Gabriel R. Numerade Educator ### Problem 10 List the first five terms of the sequence.$ a_1 = 6, a_{n+1} = \frac {a_n}{n} $GR Gabriel R. Numerade Educator ### Problem 11 List the first five terms of the sequence.$ a_1 = 2, a_{n+1} = \frac {a_n}{1 + a_n} $GR Gabriel R. Numerade Educator ### Problem 12 List the first five terms of the sequence.$ a_1 = 2, a_2 = 1, a_{a+1} = a_n - a_{n-1} $GR Gabriel R. Numerade Educator ### Problem 13 Find a formula for the general term$ 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\} $GR Gabriel R. Numerade Educator ### Problem 14 Find a formula for the general term$ a_n $of the sequence, assuming that the pattern of the first few terms continues. $$\left\{\begin{array} 4, -1, \frac {1}{4}, - \frac {1}{16}, \frac {1}{64}, . . . . .\end{array}\right\}$$ GR Gabriel R. Numerade Educator ### Problem 15 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\} $JS Joseph S. Numerade Educator ### Problem 16 Find a formula for the general term$ 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\} $GR Gabriel R. Numerade Educator ### Problem 17 Find a formula for the general term$ 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\} $GR Gabriel R. Numerade Educator ### Problem 18 Find a formula for the general term$ 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\} $GR Gabriel R. Numerade Educator ### Problem 19 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.$ a_n = \frac {3n}{1 + 6n} $GR Gabriel R. Numerade Educator ### Problem 20 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.$ a_n = 2 + \frac {(-1)^n}{n} $GR Gabriel R. Numerade Educator ### Problem 21 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.$ a_n = 1 + (- \frac {1}{2})^n $GR Gabriel R. Numerade Educator ### Problem 22 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.$ a_n = 1 + \frac{10^n}{9^n} $JS Joseph S. Numerade Educator ### Problem 23 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {3 + 5n^2}{n + n^2} $GR Gabriel R. Numerade Educator ### Problem 24 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {3 + 5n^2}{1 + n} $GR Gabriel R. Numerade Educator ### Problem 25 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {n^4}{n^3 - 2n} $GR Gabriel R. Numerade Educator ### Problem 26 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = 2 + (0.86)^n $GR Gabriel R. Numerade Educator ### Problem 27 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = 3^n 7^{-n} $GR Gabriel R. Numerade Educator ### Problem 28 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {3 \sqrt {n}}{\sqrt {n} + 2} $GR Gabriel R. Numerade Educator ### Problem 29 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = e^{-1/ \sqrt n} $GR Gabriel R. Numerade Educator ### Problem 30 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \sqrt { \frac {1 + 4n^2}{1 + n^2}} $GR Gabriel R. Numerade Educator ### Problem 31 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {4^n}{1 + 9^n} $GR Gabriel R. Numerade Educator ### Problem 32 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \cos \left( \frac {n \pi}{n + 1} \right) $GR Gabriel R. Numerade Educator ### Problem 33 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {n^2}{\sqrt {n^3 + 4n}} $GR Gabriel R. Numerade Educator ### Problem 34 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = e^{2n/(n + 2)} $GR Gabriel R. Numerade Educator ### Problem 35 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {(-1)^n}{2 \sqrt n} $GR Gabriel R. Numerade Educator ### Problem 36 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {(-1)^{n + 1}n}{n + \sqrt n} $GR Gabriel R. Numerade Educator ### Problem 37 Determine whether the sequence converges or diverges. If it converges, find the limit.$ \left \{ \frac {(2n - 1)!}{(2n + 1)!}\right \}$GR Gabriel R. Numerade Educator ### Problem 38 Determine whether the sequence converges or diverges. If it converges, find the limit.$ \left \{ \frac {\ln n}{\ln 2n} \right \} $GR Gabriel R. Numerade Educator ### Problem 39 Determine whether the sequence converges or diverges. If it converges, find the limit.$ \{ \sin n \} $GR Gabriel R. Numerade Educator ### Problem 40 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {\tan^{-1}n}{n} $GR Gabriel R. Numerade Educator ### Problem 41 Determine whether the sequence converges or diverges. If it converges, find the limit.$ \{ n^2e^{-n}\} $GR Gabriel R. Numerade Educator ### Problem 42 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \ln (n + 1) - \ln n $GR Gabriel R. Numerade Educator ### Problem 43 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac { \cos^2 n}{2^n} $GR Gabriel R. Numerade Educator ### Problem 44 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \sqrt [n]{2^{1 + 3n}} $GR Gabriel R. Numerade Educator ### Problem 45 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = n \sin (1/n) $GR Gabriel R. Numerade Educator ### Problem 46 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = 2^{-n} \cos n \pi $GR Gabriel R. Numerade Educator ### Problem 47 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \left( 1+ \frac {2}{n} \right)^n $GR Gabriel R. Numerade Educator ### Problem 48 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \sqrt[n]{n} $GR Gabriel R. Numerade Educator ### Problem 49 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \ln(2n^2 + 1) - \ln(n^2 + 1) $GR Gabriel R. Numerade Educator ### Problem 50 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac { (\ln n)^2}{n} $GR Gabriel R. Numerade Educator ### Problem 51 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \arctan (\ln n) $GR Gabriel R. Numerade Educator ### Problem 52 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = n - \sqrt {n + 1} \sqrt {n + 3} $GR Gabriel R. Numerade Educator ### Problem 53 Determine whether the sequence converges or diverges. If it converges, find the limit.$ \left \{ 0, 1, 0, 0, 1, 0, 0, 0, 1, . . . \right \} $GR Gabriel R. Numerade Educator ### Problem 54 Determine whether the sequence converges or diverges. If it converges, find the limit.$ \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 \} $GR Gabriel R. Numerade Educator ### Problem 55 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {n!}{2^n} $GR Gabriel R. Numerade Educator ### Problem 56 Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \frac {(-3)^n}{n!} $GR Gabriel R. Numerade Educator ### Problem 57 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = (-1)^n \frac {n}{n + 1} $JS Joseph S. Numerade Educator ### Problem 58 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \frac { \sin n}{n} $GR Gabriel R. Numerade Educator ### Problem 59 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \arctan \left( \frac {n^2}{n^2 + 4} \right) $GR Gabriel R. Numerade Educator ### Problem 60 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \sqrt[n]{3^n + 5^n} $GR Gabriel R. Numerade Educator ### Problem 61 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \frac {n^2 \cos n}{1 + n^2} $JS Joseph S. Numerade Educator ### Problem 62 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{n!} $JS Joseph S. Numerade Educator ### Problem 63 Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \frac {1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{(2n)^n} $JS Joseph S. Numerade Educator ### Problem 64 (a) Determine whether the sequence defined as follows is convergent or divergent:$ a_1 = 1  a_{n + 1} = 4 - a_n $for$ n \ge 1 $(b) What happens if the first term is$ a_1 = 2 $? GR Gabriel R. Numerade Educator ### Problem 65 If$ \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.

JS
Joseph S.