Problem Set - Series

## Problem Set - Series

 Define the following and give an example for each: (a)   series, (b)   converges (for a series), (c)   diverges (for a series), (d)   limit (of a series), (e)   absolutely convergent, (f)   conditionally convergent, (g)   geometric series, (h)   harmonic series, Determine if the series $\sum _{n=1}^{\infty }\frac{1}{{n}^{5}}$ converges. Use the integral test. Determine if the series $\sum _{n=1}^{\infty }\frac{1}{{n}^{2}+4}$ converges. Use the integral test. Determine if the series $\sum _{n=1}^{\infty }\frac{1}{{n}^{1/2}}$ converges. Use the integral test. Determine if the series $\sum _{n=2}^{\infty }\frac{1}{{\left(n-1\right)}^{2}}$ converges. Use the integral test. Determine if the series $\sum _{n=1}^{\infty }\frac{1}{{n}^{2}+1}$ converges. Use the comparison test. Determine if the series $\sum _{n=2}^{\infty }\frac{n}{{n}^{3}-1}$ converges. Use the comparison test. Determine if the series $\sum _{n=1}^{\infty }\frac{1}{n+1}$ converges. Use the comparison test. Determine if the series $\sum _{n=2}^{\infty }\frac{1}{n-1}$ converges. Use the comparison test. Determine if the series $\sum _{n=1}^{\infty }\frac{n}{{n}^{2}+1}$ converges. Use the comparison test. Determine if the series $\sum _{n=2}^{\infty }\frac{1}{\sqrt{n}-1}$ converges. Use the comparison test. Determine if the series $\sum _{n=1}^{\infty }\frac{2}{{3}^{n}+1}$ converges. Use the comparison test. Determine if the series $\sum _{n=1}^{\infty }\frac{{3}^{n}+1}{{4}^{n}+1}$ converges. Use the comparison test. Determine if the series $\sum _{n=1}^{\infty }\frac{{n}^{3}}{{2}^{n}}$ converges. Use the ratio test. Determine if the series $\sum _{n=1}^{\infty }\frac{n\text{!}}{{n}^{n}}$ converges. Use the ratio test. Determine if the series $\sum _{n=1}^{\infty }\frac{{2}^{n}}{n+1}$ converges. Use the ratio test. Determine if the series $\sum _{n=1}^{\infty }\frac{{2}^{n}}{n\text{!}}$ converges. Use the ratio test. Determine if the series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}}{{n}^{1/3}}$ converges. Determine if the series $\sum _{n=1}^{\infty }\sqrt{\frac{n}{n+1}}$ converges. Determine if the series $\sum _{n=1}^{\infty }\frac{1}{{n}^{7}}$ converges. Determine if the series $\sum _{n=1}^{\infty }\frac{1}{\sqrt{{n}^{2}+n}}$ converges. Determine if the series $\sum _{n=1}^{\infty }\frac{{n}^{3}}{{4}^{n}}$ converges. Determine if the series $\sum _{n=1}^{\infty }\frac{\mathrm{sin}n}{1+{n}^{2}}$ converges. Determine if the series $\frac{2}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-\cdots$ converges. Determine if the series $-\frac{1}{2}+\frac{2}{3}-\frac{3}{4}+\frac{4}{5}-\frac{5}{6}+\cdots$ converges. Determine if the series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}}{\mathrm{log}\left(n+1\right)}$ converges. Determine if the series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n}\frac{n}{{n}^{2}+1}$ converges. Determine if the series $\sum _{n=0}^{\infty }\frac{{\left(-2\right)}^{n}}{n\text{!}}$ converges absolutely. Determine if the series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n}\frac{n}{{n}^{2}+1}$ converges absolutely. Determine if the series $\sum _{n=1}^{\infty }\frac{\mathrm{cos}n}{{n}^{2}}$ converges absolutely. Determine if the series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}}{\mathrm{log}\left(n+1\right)}$ converges absolutely.