PreExam Problems: Understanding Series Nature & Convergence Radius

[csi86]

There are some problems my lecturer gave me which I can't solve ,or I am very unsure about my approach :

1. Study the nature of this series : $$\sum_{n=1}^\infty \frac{a^{n}}{n^{2}}$$

2. Expand as a series of powers of x the function $$f(x)= ln{(1+x)} + \frac {1}{1-x} + e^{2x}$$ and determine the convergence radius of the resulting one.I know that the last one is done using the Taylor but I ain't sure about my approach, some hints pls.

Thank you for your time.

 

[Office_Shredder]

For the first one, do the terms get larger or smaller as n increases?

For the second one, what is your approach that you're not sure about?

 

[csi86]

Sorry I have an error in the latex code on the first one.

@Office_Shredder : Depends on the value of a :
if it is in (-1,1) then the terms become smaller,
if if is in {-1,1} then they also become smaller,
else they become larger, as n increases.On the secound one, I know some Taylor expansions as I searched for them on wikipedia, and I know the $$e^{x}$$ expansion, as I have to get e^2x I think I will multiply that expansion by itself...

 

[csi86]

Sorry about my Latex problems... I have now finally fixed them. :D

 

[courtrigrad]

1. $$\sum_{n=1}^\infty \frac{a^{n}}{n^{2}}$$

$$\lim_{n\rightarrow \infty} \frac{a^{n}}{n^{2}}\times n = \frac{a^{n}}{n} = \infty$$ (assuming a > 1) So it converges at 1 and diverges for all other x.
2. $$f(x)= \ln{(1+x)} + \frac {1}{1-x} + e^{2x}$$

$$\ln{(1+x)} = \int \frac{1}{1+x}$$