$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

[ "article:topic", "authorname:guichard" ]

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

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

These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way.

### Exercises 11.12

Determine whether the series converges.

Ex 11.12.1    $$\sum_{n=0}^\infty {n\over n^2+4}$$ (answer)

Ex 11.12.2    $${1\over 1\cdot 2}+{1\over 3\cdot 4}+{1\over 5\cdot 6}+{1\over 7\cdot 8}+\cdots$$ (answer)

Ex 11.12.3    $$\sum_{n=0}^\infty {n\over (n^2+4)^2}$$ (answer)

Ex 11.12.4    $$\sum_{n=0}^\infty {n!\over 8^n}$$ (answer)

Ex 11.12.5     $$1-{3\over4}+{5\over8}-{7\over12}+{9\over16}+\cdots$$ (answer)

Ex 11.12.6    $$\sum_{n=0}^\infty {1\over \sqrt{n^2+4}}$$ (answer)

Ex 11.12.7    $$\sum_{n=0}^\infty {\sin^3(n)\over n^2}$$ (answer)

Ex 11.12.8    $$\sum_{n=0}^{\infty} {n\over e^n}$$ (answer)

Ex 11.12.9    $$\sum_{n=0}^\infty {n!\over 1\cdot3\cdot5\cdots(2n-1)}$$ (answer)

Ex 11.12.10   $$\sum_{n=1}^\infty {1\over n\sqrt n}$$ (answer)

Ex 11.12.11   $${1\over 2\cdot 3\cdot 4}+{2\over 3\cdot 4\cdot 5}+{3\over 4\cdot 5\cdot 6}+{4\over 5\cdot 6 \cdot 7}+\cdots$$ (answer)

Ex 11.12.12   $$\sum_{n=1}^\infty {1\cdot3\cdot5\cdots(2n-1)\over (2n)!}$$ (answer)

Ex 11.12.13   $$\sum_{n=0}^\infty {6^n\over n!}$$ (answer)

Ex 11.12.14    $$\sum_{n=1}^\infty {(-1)^{n-1}\over\sqrt n}$$ (answer)

Ex 11.12.15    $$\sum_{n=1}^\infty {2^n 3^{n-1}\over n!}$$ (answer)

Ex 11.12.16    $$1+ {5^2\over 2^2}+{5^4\over (2\cdot4)^2} +{5^6\over(2\cdot4\cdot6)^2}+ {5^8\over(2\cdot4\cdot6\cdot8)^2}+\cdots$$ (answer)

Ex 11.12.17    $$\sum_{n=1}^\infty \sin(1/n)$$ (answer)

Find the interval and radius of convergence; you need not check the endpoints of the intervals.

Ex 11.12.18    $$\sum_{n=0}^\infty {2^n\over n!}x^n$$ (answer)

Ex 11.12.19    $$\sum_{n=0}^\infty {x^n\over 1+3^n}$$ (answer)

Ex 11.12.20    $$\sum_{n=1}^\infty {x^n\over n3^n}$$ (answer)

Ex 11.12.21    $$x+{1\over 2}{x^3\over3} + {1\cdot 3\over 2\cdot4}{x^5\over5}+ {1\cdot 3\cdot5\over 2\cdot4\cdot6}{x^7\over7}+\cdots$$ (answer)

Ex 11.12.22    $$\sum_{n=1}^\infty {n!\over n^2} x^n$$ (answer)

Ex 11.12.2      $$\sum_{n=1}^\infty {(-1)^n\over n^2 3^n} x^{2n}$$ (answer)

Ex 11.12.24    $$\sum_{n=0}^\infty {(x-1)^n\over n!}$$ (answer)

Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.

Ex 11.12.25    $$2^x$$ (answer)

Ex 11.12.26     $$\ln(1+x)$$ (answer)

Ex 11.12.27     $$\ln\left({1+x\over 1-x}\right)$$ (answer)

Ex 11.12.28     $$\sqrt{1+x}$$ (answer)

Ex 11.12.29     $${1\over 1+x^2}$$ (answer)

Ex 11.12.30     $$\arctan(x)$$ (answer)

Ex 11.12.31Use the answer to the previous problem to discover a series for a well-known mathematical constant. (answer)