Problem Set - Series

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

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 ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^5}}$ converges. Use the integral test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2+4}}$ converges. Use the integral test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^{\mathrm{1/2}}}}$ converges. Use the integral test.
	Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{{(n-1)}^2}}$ converges. Use the integral test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2+1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{n}{n^3-1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n+1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{n-1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n}{n^2+1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=2}^{\infty }\frac{1}{\sqrt{n}-1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{2}{3^n+1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{3^n+1}{4^n+1}}$ converges. Use the comparison test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{2^n}}$ converges. Use the ratio test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n\text{!}}{n^n}}$ converges. Use the ratio test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{2^n}{n+1}}$ converges. Use the ratio test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{2^n}{n\text{!}}}$ converges. Use the ratio test.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{n^{\mathrm{1/3}}}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\sqrt{\frac{n}{n+1}}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^7}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{\sqrt{n^2+n}}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n^3}{4^n}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{\sin n}{1+n^2}}$ 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 ${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{\log (n+1)}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n\frac{n}{n^2+1}}$ converges.
	Determine if the series ${\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-2\right)}^n}{n\text{!}}}$ converges absolutely.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n\frac{n}{n^2+1}}$ converges absolutely.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{\cos n}{n^2}}$ converges absolutely.
	Determine if the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{\log (n+1)}}$ converges absolutely.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)