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 13.12

Determine whether the series converges.

Ex 13.12.1 $\ds\sum_{n=0}^\infty {n\over n^2+4}$ (answer)

Ex 13.12.2 $\ds {1\over 1\cdot 2}+{1\over 3\cdot 4}+{1\over 5\cdot 6}+{1\over 7\cdot 8}+\cdots$ (answer)

Ex 13.12.3 $\ds\sum_{n=0}^\infty {n\over (n^2+4)^2}$ (answer)

Ex 13.12.4 $\ds\sum_{n=0}^\infty {n!\over 8^n}$ (answer)

Ex 13.12.5 $\ds 1-{3\over4}+{5\over8}-{7\over12}+{9\over16}+\cdots$ (answer)

Ex 13.12.6 $\ds\sum_{n=0}^\infty {1\over \sqrt{n^2+4}}$ (answer)

Ex 13.12.7 $\ds\sum_{n=0}^\infty {\sin^3(n)\over n^2}$ (answer)

Ex 13.12.8 $\ds\sum_{n=0}^\infty {n\over e^n}$ (answer)

Ex 13.12.9 $\ds\sum_{n=0}^\infty {n!\over 1\cdot3\cdot5\cdots(2n-1)}$ (answer)

Ex 13.12.10 $\ds\sum_{n=1}^\infty {1\over n\sqrt n}$ (answer)

Ex 13.12.11 $\ds{1\over 2\cdot 3\cdot 4}+{2\over 3\cdot 4\cdot 5}+{3\over 4\cdot 5\cdot 6}+{4\over 5\cdot 6 \cdot 7}+\cdots$ (answer)

Ex 13.12.12 $\ds\sum_{n=1}^\infty {1\cdot3\cdot5\cdots(2n-1)\over (2n)!}$ (answer)

Ex 13.12.13 $\ds\sum_{n=0}^\infty {6^n\over n!}$ (answer)

Ex 13.12.14 $\ds\sum_{n=1}^\infty {(-1)^{n-1}\over\sqrt n}$ (answer)

Ex 13.12.15 $\ds\sum_{n=1}^\infty {2^n 3^{n-1}\over n!}$ (answer)

Ex 13.12.16 $\ds 1+ {5^2\over 2^2}+{5^4\over (2\cdot4)^2} +{5^6\over(2\cdot4\cdot6)^2}+ {5^8\over(2\cdot4\cdot6\cdot8)^2}+\cdots$ (answer)

Ex 13.12.17 $\ds \sum_{n=1}^\infty \sin(1/n)$ (answer)

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

Ex 13.12.18 $\ds\sum_{n=0}^\infty {2^n\over n!}x^n$ (answer)

Ex 13.12.19 $\ds\sum_{n=0}^\infty {x^n\over 1+3^n}$ (answer)

Ex 13.12.20 $\ds\sum_{n=1}^\infty {x^n\over n3^n}$ (answer)

Ex 13.12.21 $\ds x+{1\over 2}{x^3\over3} + {1\cdot 3\over 2\cdot4}{x^5\over5}+ {1\cdot 3\cdot5\over 2\cdot4\cdot6}{x^7\over7}+\cdots$ (answer)

Ex 13.12.22 $\ds\sum_{n=1}^\infty {n!\over n^2} x^n$ (answer)

Ex 13.12.23 $\ds\sum_{n=1}^\infty {(-1)^n\over n^2 3^n} x^{2n}$ (answer)

Ex 13.12.24 $\ds\sum_{n=0}^\infty {(x-1)^n\over n!}$ (answer)

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

Ex 13.12.25 $\ds 2^x$ (answer)

Ex 13.12.26 $\ds \ln(1+x)$ (answer)

Ex 13.12.27 $\ds \ln\left({1+x\over 1-x}\right)$ (answer)

Ex 13.12.28 $\ds \sqrt{1+x}$ (answer)

Ex 13.12.29 $\ds {1\over 1+x^2}$ (answer)

Ex 13.12.30 $\ds \arctan(x)$ (answer)

Ex 13.12.31 Use the answer to the previous problem to discover a series for a well-known mathematical constant. (answer)