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anonymous
 5 years ago
Show the Series n=1 to infinity nX^n converges to f(x)=x/(1x)^2... without the use of taylors theorem
anonymous
 5 years ago
Show the Series n=1 to infinity nX^n converges to f(x)=x/(1x)^2... without the use of taylors theorem

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0take the derivative of \[\frac{1}{1x}=\Sigma x^n\] to get \[\frac{1}{(1x)^2}=\Sigma n x^{n1}\]

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

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0of course you have to be able to justify taking the derivative term by term, and also note that the radius of convergence is 1 < x < 1

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Thanks Yeah I did the term by term derivative, so is the initial geo series starting at n=1?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the initial series starts at n = 0 not n = 1, but this is not a problem because the final one gives 0 for n = 0 so you might as well start at n = 1
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