TT1 Problem 2 (night)

[Victor Ivrii]

Determine the radius of convergence

(a)  $\displaystyle{\sum_{n=1}^\infty \frac{z^n}{2^n n^2}}$

(b) $\displaystyle{\sum_{n=1}^\infty  \frac{z^{3n} (3n)!}{20^n (2n)! }}$

If the radius of convergence is $R$, $0<R< \infty$, determine for each  $z\colon |z|=R$ if this series converges.

[Heng Kan]

See the attached scanned picture.

[Xiting Kuang]

Just a concern, it says in the problem that R should be positive.

[Heng Kan]

I think the question means that if the radius of convergence is positive,you have to figure out whether the series is convergent at the radius of convergence. It doesn't mean the radius is always positive.

[Victor Ivrii]

I think the question means that if the radius of convergence is positive,you have to figure out whether the series is convergent at the radius of convergence. It doesn't mean the radius is always positive.

Indeed.