# 11: Sequences and Series (Exercises)

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These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

11.1: Sequences

**Ex 11.1.1** Compute \(\lim_{x\to\infty} x^{1/x}\). (answer)

**Ex 11.1.2** Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\).

**Ex 11.1.3** Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. If it converges, compute the limit. (answer)

**Ex 11.1.4** Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. If it converges, compute the limit. (answer)

**Ex 11.1.5** Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. If it converges, compute the limit. (answer)

**Ex 11.1.6** Determine whether \(\left\{{2^n\over n!}\right\}_{n=0}^{\infty}\) converges or diverges. (answer)

## 11.2: Series

**Ex 11.2.1** Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. (answer)

**Ex 11.2.2** Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. (answer)

**Ex 11.2.3** Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. (answer)

**Ex 11.2.4** Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). (answer)

**Ex 11.2.5** Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). (answer)

**Ex 11.2.6** Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). (answer)

**Ex 11.2.7** Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). (answer)

**Ex 11.2.8** Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). (answer)

**Ex 11.2.9** Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). (answer)

## 11.3: The Integral Test

Determine whether each series converges or diverges.

**Ex 11.3.1** \(\sum_{n=1}^\infty {1\over n^{\pi/4}}\) (answer)

**Ex 11.3.2** \(\sum_{n=1}^\infty {n\over n^2+1}\) (answer)

**Ex 11.3.3** \(\sum_{n=1}^\infty {\ln n\over n^2}\) (answer)

**Ex 11.3.4** \(\sum_{n=1}^\infty {1\over n^2+1}\) (answer)

**Ex 11.3.5** \(\sum_{n=1}^\infty {1\over e^n}\) (answer)

**Ex 11.3.6** \(\sum_{n=1}^\infty {n\over e^n}\) (answer)

**Ex 11.3.7** \(\sum_{n=2}^\infty {1\over n\ln n}\) (answer)

**Ex 11.3.8** \(\sum_{n=2}^\infty {1\over n(\ln n)^2}\) (answer)

**Ex 11.3.9** Find an \(N\) so that \(\sum_{n=1}^\infty {1\over n^4}\) is between \(\sum_{n=1}^N {1\over n^4}\) and \(\sum_{n=1}^N {1\over n^4} + 0.005\). (answer)

**Ex 11.3.10** Find an \(N\) so that \(\sum_{n=0}^\infty {1\over e^n}\) is between \(\sum_{n=0}^N {1\over e^n}\) and \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). (answer)

**Ex 11.3.11** Find an \(N\) so that \(\sum_{n=1}^\infty {\ln n\over n^2}\) is between \(\sum_{n=1}^N {\ln n\over n^2}\) and \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). (answer)

**Ex 11.3.12** Find an \(N\) so that \(\sum_{n=2}^\infty {1\over n(\ln n)^2}\) is between \(\sum_{n=2}^N {1\over n(\ln n)^2}\) and \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). (answer)

## 11.4: Alternating Series

Determine whether the following series converge or diverge.

**Ex 11.4.1** \(\sum_{n=1}^\infty {(-1)^{n-1}\over 2n+5}\) (answer)

**Ex 11.4.2** \(\sum_{n=4}^\infty {(-1)^{n-1}\over \sqrt{n-3}}\) (answer)

**Ex 11.4.3** \(\sum_{n=1}^\infty (-1)^{n-1}{n\over 3n-2}\) (answer)

**Ex 11.4.4** \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer)

**Ex 11.4.5** Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^3}\) to two decimal places. (answer)

**Ex 11.4.6 **Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. (answer)

## 11.5: Comparison Test

Determine whether the series converge or diverge.

**Ex 11.5.1 **\(\sum_{n=1}^\infty {1\over 2n^2+3n+5} \) (answer)

**Ex 11.5.2 **\(\sum_{n=2}^\infty {1\over 2n^2+3n-5} \) (answer)

**Ex 11.5.3 **\(\sum_{n=1}^\infty {1\over 2n^2-3n-5} \) (answer)

**Ex 11.5.4 **\(\sum_{n=1}^\infty {3n+4\over 2n^2+3n+5} \) (answer)

**Ex 11.5.5 **\(\sum_{n=1}^\infty {3n^2+4\over 2n^2+3n+5} \) (answer)

**Ex 11.5.6 **\(\sum_{n=1}^\infty {\ln n\over n}\) (answer)

**Ex 11.5.7 **\(\sum_{n=1}^\infty {\ln n\over n^3}\) (answer)

**Ex 11.5.8 **\(\sum_{n=2}^\infty {1\over \ln n}\) (answer)

**Ex 11.5.9 **\(\sum_{n=1}^\infty {3^n\over 2^n+5^n}\) (answer)

**Ex 11.5.10 **\(\sum_{n=1}^\infty {3^n\over 2^n+3^n}\) (answer)

## 11.6: Absolute Convergence

Determine whether each series converges absolutely, converges conditionally, or diverges.

**Ex 11.6.1 **\(\sum_{n=1}^\infty (-1)^{n-1}{1\over 2n^2+3n+5}\) (answer)

**Ex 11.6.2** \(\sum_{n=1}^\infty (-1)^{n-1}{3n^2+4\over 2n^2+3n+5}\) (answer)

**Ex 11.6.3** \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer)

**Ex 11.6.4** \(\sum_{n=1}^\infty (-1)^{n-1} {\ln n\over n^3}\) (answer)

**Ex 11.6.5** \(\sum_{n=2}^\infty (-1)^n{1\over \ln n}\) (answer)

**Ex 11.6.6** \(\sum_{n=0}^\infty (-1)^{n} {3^n\over 2^n+5^n}\) (answer)

**Ex 11.6.7** \(\sum_{n=0}^\infty (-1)^{n} {3^n\over 2^n+3^n}\) (answer)

**Ex 11.6.8** \(\sum_{n=1}^\infty (-1)^{n-1} {\arctan n\over n}\) (answer)

## 11.7: The Ratio and Root Tests

**Ex 11.7.1 **Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\).

**Ex 11.7.2 **Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\).

**Ex 11.7.3 **Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\).

**Ex 11.7.4 **Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\).

Determine whether the series converge.

**Ex 11.7.5 **\(\sum_{n=0}^\infty (-1)^{n}{3^n\over 5^n}\) (answer)

**Ex 11.7.6 **\(\sum_{n=1}^\infty {n!\over n^n}\) (answer)

**Ex 11.7.7 **\(\sum_{n=1}^\infty {n^5\over n^n}\) (answer)

**Ex 11.7.8 **\(\sum_{n=1}^\infty {(n!)^2\over n^n}\) (answer)

**Ex 11.7.9 **Prove theorem __11.7.3__, the root test.

## 11.8: Power Series

Find the radius and interval of convergence for each series. In exercises __3__ and __4__, do not attempt to determine whether the endpoints are in the interval of convergence.

**Ex 11.8.1** \(\sum_{n=0}^\infty n x^n\) (answer)

**Ex 11.8.2 **\(\sum_{n=0}^\infty {x^n\over n!}\) (answer)

**Ex 11.8.3 **\(\sum_{n=1}^\infty {n!\over n^n}x^n\) (answer)

**Ex 11.8.4** \(\sum_{n=1}^\infty {n!\over n^n}(x-2)^n\) (answer)

**Ex 11.8.5 **\(\sum_{n=1}^\infty {(n!)^2\over n^n}(x-2)^n\) (answer)

**Ex 11.8.6 **\(\sum_{n=1}^\infty {(x+5)^n\over n(n+1)}\) (answer)

## 11.9: Calculus with Power Series

**Ex 11.9.1 **Find a series representation for \(\ln 2\). (answer)

**Ex 11.9.2** Find a power series representation for \(1/(1-x)^2\). (answer)

**Ex 11.9.3 **Find a power series representation for \( 2/(1-x)^3\). (answer)

**Ex 11.9.4** Find a power series representation for \( 1/(1-x)^3\). What is the radius of convergence? (answer)

**Ex 11.9.5** Find a power series representation for \(\int\ln(1-x)\,dx\). (answer).

## 11.10: Taylor Series

For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence.

**Ex 11.10.1 **\(\cos x\) (answer)

**Ex 11.10.2 **\( e^x\) (answer)

**Ex 11.10.3 **\(1/x\), \(a=5\) (answer)

**Ex 11.10.4 **\(\ln x\), \(a=1\) (answer)

**Ex 11.10.5 **\(\ln x\), \(a=2\) (answer)

**Ex 11.10.6 **\( 1/x^2\), \(a=1\) (answer)

**Ex 11.10.7 **\(1/\sqrt{1-x}\) (answer)

**Ex 11.10.8 **Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). (answer)

**Ex 11.10.9 **Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). (answer)

**Ex 11.10.10 **Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). (answer)

## 11.11: Taylor's Theorem

**Ex 11.11.1** Find a polynomial approximation for \(\cos x\) on \([0,\pi]\), accurate to \( \pm 10^{-3}\) (answer)

**Ex 11.11.2** How many terms of the series for \(\ln x\) centered at 1 are required so that the guaranteed error on \([1/2,3/2]\) is at most \( 10^{-3}\)? What if the interval is instead \([1,3/2]\)? (answer)

**Ex 11.11.3 **Find the first three nonzero terms in the Taylor series for \(\tan x\) on \([-\pi/4,\pi/4]\), and compute the guaranteed error term as given by Taylor's theorem. (You may want to use Sage or a similar aid.) (answer)

**Ex 11.11.4** Show that \(\cos x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as N approaches infinity.

**Ex 11.11.5** Show that \(e^x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as \(N\) approaches infinity.