Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

11: Sequences and Series (Exercises)

 

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

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.