# 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.