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anonymous

  • 5 years ago

Determine a function which is represented by that summation given below

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  1. anonymous
    • 5 years ago
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    \[\sum _{n=1}^{\infty } \text{nx}^n\]

  2. anonymous
    • 5 years ago
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    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.

  3. anonymous
    • 5 years ago
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    \[\Sigma_1^\infty x^n=\frac{1}{1-x}\] for -1<x<1 yes?

  4. anonymous
    • 5 years ago
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    yes

  5. anonymous
    • 5 years ago
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    taking derivatives we get \[\Sigma_1^\infty nx^{n-1}=\frac{1}{(1-x)^2}\]

  6. anonymous
    • 5 years ago
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    okay

  7. anonymous
    • 5 years ago
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    multiply both sides by x to get \[\Sigma_1^\infty nx^n=\frac{x}{(1-x)^2}\]

  8. anonymous
    • 5 years ago
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    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.

  9. anonymous
    • 5 years ago
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    Okay,thanks

  10. anonymous
    • 5 years ago
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    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.

  11. anonymous
    • 5 years ago
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    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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