Find the interval of convergence and sum of the series
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.
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.)
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.
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?
Yes; that is the right way to go.