Series practice list

Main list

Determine whether each of the series is absolutely convergent, conditionally convergent, or divergent, using any convergence test:

(1) ${\displaystyle \sum _{n=1}^{\infty }(1+\frac{1}{n})2^{-n}}$ $$ (2) ${\displaystyle \sum _{n=1}^{\infty }\sin \left(\left(\frac{2n+1}{2}\right)\pi \right)n^{-1/2}}$ $$ (3) ${\displaystyle \sum _{n=1}^{\infty }\frac{\ln n}{n+2}}$

(4) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\ln \left(\frac{n}{3n+1}\right)}$ $$ (5) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-2)}^n}{n^2}}$ $$ (6) ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{1}{9}\right)}^n(2n)!}$

(7) ${\displaystyle \sum _{n=1}^{\infty }\frac{1\cdot 3\cdot 5\cdots (2n-1)}{5^{n}n!}}$ $$ (8) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^n}{3n}}$ $$ (9) ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{-1}{3n}\right)}^n}$

(10) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\frac{n^{2n}}{{(1+2n^2)}^n}}$ $$ (11) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\frac{n}{n^4+1}}$ $$ (12) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\frac{\ln n}{\sqrt{n}}}$

(13) ${\displaystyle \sum _{n=1}^{\infty }\frac{\ln n}{n^2}}$

Extra - Rogawski problems

The following problems are drawn from the J. Rogawski textbook, p585:

For each series, state a convergence test that will show whether it converges or diverges. (If you have time, write further details for application of the test.)

(43) ${\displaystyle \sum _{n=1}^{\infty }\frac{2^n+4^n}{7^n}}$ $$ (44) ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{n!}}$ $$ (45) ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{5^n}}$

(46) ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{n{(\ln n)}^3}}$ $$ (47) ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{\sqrt{n^3-n^2}}}$ $$ (48) ${\displaystyle \sum _{n=1}^{\infty }\frac{n^2+4n}{3n^4+9}}$

(49) ${\displaystyle \sum _{n=1}^{\infty }{n}^{-0.8}}$ $$ (50) ${\displaystyle \sum _{n=1}^{\infty }{(0.8)}^{-n}{n}^{-0.8}}$ $$ (51) ${\displaystyle \sum _{n=1}^{\infty }{4}^{-2n+1}}$

(52) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-1)}^{n-1}}{\sqrt{n}}}$ $$ (53) ${\displaystyle \sum _{n=1}^{\infty }\sin \frac{1}{n^2}}$ $$ (54) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\cos \frac{1}{n}}$

(55) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-2)}^n}{\sqrt{n}}}$ $$ (56) ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{n}{n+12}\right)}^n}$