Radius of Convergence for Power Series

[Wesyu]

Anyone knows how to do these?

 

http://www.math.mcgill.ca/jakobson/courses/ma262/a1.pdf

[bloodhound]

havent done this for a while . but here goes for the first one.

 

use the ratio test

 

see if the limit as n tends to infinity of

 

[math]|a_{n+1}/a_{n}|[/math] exists. where in this case.

 

[math]a_{n}=x^{n}\sum_{i=1}^{n}\frac{1}{i}[/math]

 

so we have [math]|a_{n+1}/a_{n}|=|x\frac{\sum_{i=1}^{n+1}1/i}{\sum_{i=1}^{n}1/i}|[/math]

 

which just converges to |x| as n tends to infinity

 

therefore the series converges if |x|<1

 

i.e -1<x<1.

 

when x = -1 . the series doesnt converge as [math]\sum_{i=1}^{n}1/i[/math] doesnt converge to 0. and if x =1 the seires doesnt converge as it is the standard harmonic series.

[bloodhound]

have to think a lot harder for the second one. its quite easy to show that it does indeed converge for some x . but what i dont know. it does defintely converge if

 

-1/10 < x < 1/10 . but that is the radius of convergence for the comaprision series i used to check if the original series converges.

[bloodhound]

for the comparision for the second question , i used . [math]\sum_{n=1}^{\infty}\frac{10^{n}x^{n}}{n}[/math]