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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ex 11.12.23 $\ds\sum_{n=1}^\infty {(-1)^n\over n^2 3^n} x^{2n}$ (answer)

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

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

Ex 11.12.25 $\ds 2^x$ (answer)

Ex 11.12.26 $\ds \ln(1+x)$ (answer)

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

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

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

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

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