Sum and interval of convergence

: Math 2730 Sequences and Series: Sum and interval of convergence
   By Martin Daniels on Tuesday, April 01, 2008 - 10:40 pm: Edit Post

Find the interval of convergence and sum of the series

Sigma(1,inf) nx^n

I don't know where to start this one. It seems like maybe I should relate it to one of the known Maclaurin series but I just can't seem to find something appropriate.


   By Sasho on Tuesday, April 01, 2008 - 11:14 pm: Edit Post

For the radius of convergence you may use, say, the ratio test.
In order to find the sum, here is one way to start:
Sigma(1,inf) nxn = x Sigma(1,inf) nxn-1
(It should now be relatively easy to relate the sum to the right with Sigma(0,inf) xn.)


   By Martin Daniels on Thursday, April 03, 2008 - 01:02 pm: Edit Post

Yes, i see now that Sigma(1,inf)nx^(n-1) is a geometric series.

The n in front of the x^(n-1) is really confusing me though. I don't know how to include/exclude it when writing the sum.


   By Martin Daniels on Thursday, April 03, 2008 - 01:15 pm: Edit Post

Actually, now I'm starting to see it.

The n comes from differentiating x^n and then clearly the x^(n-1) comes from the same place.

So then d/dx [Sigma(0,inf) x^n] = Sigma(1,inf) nx(n-1)

Its sum would then be 1/(1-x)^2 for |x|< 1

On the right track?


   By Sasho on Thursday, April 03, 2008 - 01:59 pm: Edit Post

Yes; that is the right way to go.


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