Determine a function which is represented by that summation given below

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Determine a function which is represented by that summation given below

Mathematics
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\[\sum _{n=1}^{\infty } \text{nx}^n\]
ok i guess we will worry about the radius of convergence second. this looks almost like a derivative, so we will treat it as one.
\[\Sigma_1^\infty x^n=\frac{1}{1-x}\] for -1

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yes
taking derivatives we get \[\Sigma_1^\infty nx^{n-1}=\frac{1}{(1-x)^2}\]
okay
multiply both sides by x to get \[\Sigma_1^\infty nx^n=\frac{x}{(1-x)^2}\]
i have been very sloppy here, especially from where we started. i think we have to be careful with starting at n = 0 or n = 1, but i will let you worry about this. i will also let you worry about the validity of differentiating a power series term by term and the radius of convergence. but the general idea is there and i think the answer is correct modulo those details.
Okay,thanks
i think in fact the very first line i wrote was incorrect. i think \[\Sigma_0^\infty x^n = \frac{1}{1-x}\] no what i wrote.
but of course when n = 0 in your power series you get 0 anyway, so you might as well start at 1!

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