Problem Set - Power 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

Power series

	Write out the first four terms of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
	Write out the first four terms of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
	Write out the first four terms of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(x-1)}^n}{n}}$.
	Write out the first four terms of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^n}{n+2}}$.
	Find the Taylor expansion of $e^x$ at $x=0$.
	Find the Taylor expansion of $\sinh x$ at $x=0$.
	Find the Taylor expansion of $\frac{1}{1-x}$ at $x=0$.
	Find the Taylor expansion of $e^x$ at $x=2$.
	Find the Taylor expansion of $\log x$ at $x=1$.
	Find the Taylor expansion of $\frac{1}{x^2}$ at $x=1$.
	Prove the identity $e^{ix}=\cos x+i\sin x$.
	Prove the identity $e^x=\cosh x+\sinh x$.
	Find the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
	Find the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{{(x+1)}^n}{{(n+1)}^2}}$.
	Find the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
	Find the radius of convergence of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(2x-1)}^n}{\sqrt[3]{n}}}$.
	Find the interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n+1}}$.
	Find the interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{{(x+1)}^n}{{(n+1)}^2}}$.
	Find the interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n\text{!}}}$.
	Find the interval of convergence of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{{(2x-1)}^n}{\sqrt[3]{n}}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }n{x}^{n-1}}$.
	Find the sum of the series ${\displaystyle \sum _{n=0}^{\infty }\frac{x^{n+1}}{n+1}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{n}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{n}{3^{n-1}}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n{2}^{n+1}}}$.
	Find the sum of the series ${\displaystyle \sum _{n=1}^{\infty }n(n-1){\left(\frac{1}{4}\right)}^n}$.
	Find the power series representation of ${\displaystyle \frac{1}{1+2x}}$ and determine its radius of convergence.
	Find the power series representation of ${\displaystyle \frac{1}{1+x^2}}$ and determine its radius of convergence.
	Find the power series representation of ${\displaystyle \frac{x}{1+x}}$ and determine its radius of convergence.
	Find the power series representation of ${\displaystyle \frac{1}{{(1+x)}^2}}$ and determine its radius of convergence.
	Find the power series representation of $\arctan x$ and determine its radius of convergence.
	Find the power series representation of $\log (2+x)$ and determine its radius of convergence.
	Find the power series representation of $\int e^{x^3}dx$.
	Find the power series representation of ${\displaystyle \int \frac{\sinh x}{x}dx}$.
	Find an infinite series representation of ${\displaystyle {\int }_{-1}^1\frac{\sinh x}{x}dx}$.
	Find an infinite series representation of ${\displaystyle {\int }_0^1{e}^{x^3}dx}$.

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)