微積分收斂半徑問題

find the radius of convergence and sum of the series

Σ(k=n~無限大) (x^n)/n=x+x^2/2+x^3/3.......

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  • 1 decade ago
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    (1) Radius of convergence:

    ratio test: lim(n->∞) | an+1/ an | <1 => convergent

    =>lim(n->∞) | [xn+1*n] / [xn*(n+1)] | = |x| <1

    故Radius of convergence=1 (收歛區間為 -1<=x<1, 收歛半徑=收歛寬度一半)

    (2)設收歛函數(總和)為f(x), 則

    f'(x)=1+x+x2+x3+... (逐項求導函數)

    =1/(1-x) (無窮等比)

    (積分)=> f(x)= - ln |1-x| , -1<=x<1

    Source(s): me
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