anonymous
  • anonymous
Determine a function which is represented by that summation given below
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
\[\sum _{n=1}^{\infty } \text{nx}^n\]
anonymous
  • anonymous
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.
anonymous
  • anonymous
\[\Sigma_1^\infty x^n=\frac{1}{1-x}\] for -1

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anonymous
  • anonymous
yes
anonymous
  • anonymous
taking derivatives we get \[\Sigma_1^\infty nx^{n-1}=\frac{1}{(1-x)^2}\]
anonymous
  • anonymous
okay
anonymous
  • anonymous
multiply both sides by x to get \[\Sigma_1^\infty nx^n=\frac{x}{(1-x)^2}\]
anonymous
  • anonymous
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.
anonymous
  • anonymous
Okay,thanks
anonymous
  • anonymous
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.
anonymous
  • anonymous
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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