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anonymous
 5 years ago
Determine a function which is represented by that summation given below
anonymous
 5 years ago
Determine a function which is represented by that summation given below

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\sum _{n=1}^{\infty } \text{nx}^n\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ok i guess we will worry about the radius of convergence second. this looks almost like a derivative, so we will treat it as one.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\Sigma_1^\infty x^n=\frac{1}{1x}\] for 1<x<1 yes?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0taking derivatives we get \[\Sigma_1^\infty nx^{n1}=\frac{1}{(1x)^2}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0multiply both sides by x to get \[\Sigma_1^\infty nx^n=\frac{x}{(1x)^2}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i think in fact the very first line i wrote was incorrect. i think \[\Sigma_0^\infty x^n = \frac{1}{1x}\] no what i wrote.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0but 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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