11: Sequences and Series (Exercises)

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.

11.12: Additional Exercises