# Toronto Math Forum

## MAT334-2018F => MAT334--Tests => Term Test 1 => Topic started by: Victor Ivrii on October 19, 2018, 04:13:20 AM

Title: TT1 Problem 2 (night)
Post by: Victor Ivrii on October 19, 2018, 04:13:20 AM

(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.
Title: Re: TT1 Problem 2 (night)
Post by: Heng Kan on October 19, 2018, 09:34:01 AM
See the attached scanned picture.
Title: Re: TT1 Problem 2 (night)
Post by: Xiting Kuang on October 19, 2018, 09:37:28 AM
Just a concern, it says in the problem that R should be positive.
Title: Re: TT1 Problem 2 (night)
Post by: Heng Kan on October 19, 2018, 09:45:26 AM
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.
Title: Re: TT1 Problem 2 (night)
Post by: Victor Ivrii on October 20, 2018, 03:20:20 PM
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.