Problem Set - Power series

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 ∑n=0∞xnn+1.

Write out the first four terms of the series ∑n=0∞xnn!.

Write out the first four terms of the series ∑n=1∞(x-1)nn.

Write out the first four terms of the series ∑n=0∞(-1)nxnn+2.

Find the Taylor expansion of ex at x=0.

Find the Taylor expansion of sinhx at x=0.

Find the Taylor expansion of 11-x at x=0.

Find the Taylor expansion of ex at x=2.

Find the Taylor expansion of logx at x=1.

Find the Taylor expansion of 1x2 at x=1.

Prove the identity eix=cosx+isinx.

Prove the identity ex=coshx+sinhx.

Find the radius of convergence of the series ∑n=0∞xnn+1.

Find the radius of convergence of the series ∑n=0∞(-1)n(x+1)n(n+1)2.

Find the radius of convergence of the series ∑n=0∞xnn!.

Find the radius of convergence of the series ∑n=1∞(2x-1)nn3.

Find the interval of convergence of the series ∑n=0∞xnn+1.

Find the interval of convergence of the series ∑n=0∞(-1)n(x+1)n(n+1)2.

Find the interval of convergence of the series ∑n=0∞xnn!.

Find the interval of convergence of the series ∑n=1∞(2x-1)nn3.

Find the sum of the series ∑n=1∞nxn-1.

Find the sum of the series ∑n=0∞xn+1n+1.

Find the sum of the series ∑n=1∞xnn.

Find the sum of the series ∑n=1∞n3n-1.

Find the sum of the series ∑n=1∞1n2n+1.

Find the sum of the series ∑n=1∞n(n-1)(14)n.

Find the power series representation of 11+2x and determine its radius of convergence.

Find the power series representation of 11+x2 and determine its radius of convergence.

Find the power series representation of x1+x and determine its radius of convergence.

Find the power series representation of 1(1+x)2 and determine its radius of convergence.

Find the power series representation of arctanx 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 ∫ex3dx.

Find the power series representation of ∫sinhxxdx.

Find an infinite series representation of ∫-11sinhxxdx.

Find an infinite series representation of ∫01ex3dx.

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)