Does it converge, and which test proves it

Convergence questions are really test-selection questions. Each page gives the verdict, names the ONE test that settles it, and says why the tempting alternatives come back inconclusive. Of the 41 series here, 30 converge.

• $\sum_{n=1}^{\infty}\frac{1}{n}$divergesp-series
• $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$convergesp-series
• $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$divergesp-series
• $\sum_{n=1}^{\infty}\frac{1}{n^{3}}$convergesp-series
• $\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}$convergesp-series
• $\sum_{n=1}^{\infty}\frac{1}{2^{n}}$convergesgeometric
• $\sum_{n=1}^{\infty}\frac{1}{3^{n}}$convergesgeometric
• $\sum_{n=1}^{\infty}\frac{3^{n}}{4^{n}}$convergesgeometric
• $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2^{n}}$convergesgeometric
• $\sum_{n=1}^{\infty}\frac{n}{n+1}$divergesnth-term
• $\sum_{n=1}^{\infty}\frac{n^{2}}{2n^{2}+1}$divergesnth-term
• $\sum_{n=1}^{\infty}\frac{2^{n}}{n^{2}}$divergesratio
• $\sum_{n=1}^{\infty}\frac{n}{2^{n}}$convergesratio
• $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}$convergesalternating
• $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}$convergesalternating
• $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n}}$convergesalternating
• $\sum_{n=2}^{\infty}\frac{1}{n\ln n}$divergesintegral
• $\sum_{n=1}^{\infty}\frac{1}{n(n+1)}$convergestelescoping
• $\sum_{n=1}^{\infty}\frac{\sin n}{n^{2}}$convergescomparison
• $\sum_{n=1}^{\infty}\frac{1}{n^{2}+1}$convergeslimit-comparison
• $\sum_{n=2}^{\infty}\frac{1}{\ln n}$divergescomparison
• $\sum_{n=1}^{\infty}\frac{1}{n!}$convergesratio
• $\sum_{n=1}^{\infty}\frac{2^{n}}{n!}$convergesratio
• $\sum_{n=1}^{\infty}\frac{3^{n}}{n!}$convergesratio
• $\sum_{n=1}^{\infty}\frac{n^{2}}{n!}$convergesratio
• $\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$convergesratio
• $\sum_{n=1}^{\infty}\frac{n^{n}}{n!}$divergesratio
• $\sum_{n=1}^{\infty}\frac{(n!)^{2}}{(2n)!}$convergesratio
• $\sum_{n=1}^{\infty}\frac{1}{n^{n}}$convergesroot
• $\sum_{n=1}^{\infty}\frac{n}{3^{n}}$convergesratio
• $\sum_{n=1}^{\infty}\frac{2}{3^{n}}$convergesgeometric
• $\sum_{n=1}^{\infty}\frac{2^{n}}{5^{n}}$convergesgeometric
• $\sum_{n=2}^{\infty}\frac{1}{n^{2}-1}$convergeslimit-comparison
• $\sum_{n=1}^{\infty}\frac{n}{n^{2}+1}$divergeslimit-comparison
• $\sum_{n=1}^{\infty}\frac{1}{2n+1}$divergeslimit-comparison
• $\sum_{n=1}^{\infty}\frac{\sqrt{n}}{n^{2}+1}$convergeslimit-comparison
• $\sum_{n=1}^{\infty}\frac{n}{n^{3}+1}$convergeslimit-comparison
• $\sum_{n=1}^{\infty}\frac{1}{n+\sqrt{n}}$divergeslimit-comparison
• $\sum_{n=1}^{\infty}\frac{\arctan n}{n^{2}}$convergescomparison
• $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n+1}$convergesalternating
• $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n+1}$convergesalternating