Few question related to power series

[seto6]

Homework Statement

let a_n= $$\sum^{k=1}_{n}$$ 1/$$\sqrt{k}$$
what is the radius of convergence of $$\Sigma$$$$\suma^{n=1}_{infinity} a_{n}x^n$$


i tired including the a_n term into the x^n equation then i got stuck.. help please



2. Suppose that $$\alpha$$ and $$\beta$$ are positive real numbers with $$\alpha$$ < $$\beta$$. find a power series with an interval of convergence that is of the given interval:

I. ($$\alpha$$,$$\beta$$)
II. [$$\alpha$$,$$\beta$$)

i basically came up with power series that i know that has this convergence, but is there a systematic way of doing it, with a real proof.

Thank you in advance

 

[lanedance]

do you know about the harmonc series $$\sum \frac{1}{k}$$ and whether it converges?

could you compare your seres to it?

 

[seto6]

i don't think it would help much

 

[lanedance]

ok but you know a_n diverges as n gets large right?

have you tried a ratio test?