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Math Help - find the sum of series

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    find the sum of series

    starting with geometric series sigma from n=0 to infinity of x^n find the sum of the series of sigma from n= 1 to infinity of n(x)^(n-1) lxl<1

    b) find the sum of each of the following series,
    1) sigma from n = 1 to infinity nX^n, lxl<1
    2) sigma from n=1 to infity of n/2^n
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    Quote Originally Posted by twilightstr View Post
    starting with geometric series sigma from n=0 to infinity of x^n find the sum of the series of sigma from n= 1 to infinity of n(x)^(n-1) lxl<1

    b) find the sum of each of the following series,
    1) sigma from n=1 to infinity nX^n, lxl<1
    2) sigma from n=1 to infity of n/2^n
    A. Let \ell(x):=\textstyle\sum\nolimits_{n=0}^{\infty}x^{  n}, and show that \ell(x)=1/(1-x), then calculate \ell^{\prime}(x) for |x|<1.
    B.1. Calculate x\ell^{\prime}(x) for |x|<1.
    B.2. Check \ell^{\prime}(1/2)/2.
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