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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.

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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) nx^{n} = x Sigma(1,inf) nx^{n-1}

(It should now be relatively easy to relate the sum to the right with Sigma(0,inf) x^{n}.)

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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.

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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?

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Yes; that is the right way to go.