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anonymous

  • 5 years ago

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

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  1. anonymous
    • 5 years ago
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    take the derivative of \[\frac{1}{1-x}=\Sigma x^n\] to get \[\frac{1}{(1-x)^2}=\Sigma n x^{n-1}\]

  2. anonymous
    • 5 years ago
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    then multiply by x to get \[\frac{x}{(1-x)^2}=\Sigma nx^n\]

  3. anonymous
    • 5 years ago
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    of 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

  4. anonymous
    • 5 years ago
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    Thanks Yeah I did the term by term derivative, so is the initial geo series starting at n=1?

  5. anonymous
    • 5 years ago
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    the 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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