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Math Help - Power Series #2

  1. #1
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    Power Series #2

    This is the last of the set of real analysis questions. Any help would be greatly appreciated:

    A series  \sum_{n=0}^{\infty}{{a_n}} is said to be Abel-summable to L if the power series
    f(x) =  \sum_{n=0}^{\infty} a_n*x^n converges for all x Є [0,1) and L = lim f(x) as x approaches 1 from the negative side.

    a. Show that any series that converges to a limit L is also Abel-summable to L.
    b. Show that  \sum_{n=0}^{\infty} (-1)^n is Abel-summable and find the sum.
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  2. #2
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    Quote Originally Posted by ajj86 View Post

    A series  \sum_{n=0}^{\infty}{{a_n}} is said to be Abel-summable to L if the power series
    f(x) =  \sum_{n=0}^{\infty} a_n*x^n converges for all x Є [0,1) and L = lim f(x) as x approaches 1 from the negative side.

    a. Show that any series that converges to a limit L is also Abel-summable to L.
    since \sum a_n is convergent, the interval of convergence of the power series \sum a_nx^n contains 1. thus the radius of convergence is at least 1, i.e. it converges at least over the interval (-1,1].


    b. Show that  \sum_{n=0}^{\infty} (-1)^n is Abel-summable and find the sum.
    suppose |x| < 1. then \sum_{n \geq 0} (-1)^n x^n = \sum_{n \geq 0} (-x)^n = \frac{1}{1+x}. therefore L=\lim_{x\to1^{-}} \sum_{n \geq 0}(-1)^n x^n = \frac{1}{2}.
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