### Author Topic: TT1 Problem 2 (night)  (Read 6373 times)

#### Victor Ivrii

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##### TT1 Problem 2 (night)
« 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.

#### Heng Kan

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##### Re: TT1 Problem 2 (night)
« Reply #1 on: October 19, 2018, 09:34:01 AM »
See the attached scanned picture.

#### Xiting Kuang

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##### Re: TT1 Problem 2 (night)
« Reply #2 on: October 19, 2018, 09:37:28 AM »
Just a concern, it says in the problem that R should be positive.

#### Heng Kan

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##### Re: TT1 Problem 2 (night)
« Reply #3 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.

#### Victor Ivrii

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##### Re: TT1 Problem 2 (night)
« Reply #4 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.