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Math Help - power series justification

  1. #1
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    power series justification

    Hi.

    Quick question.
    Given a power series of the form:

    sigma cn(x+3)^n. It is given that this series converges when x=0 and diverges when
    x = -8

    Now does it converge when the series is

    sigma cn2^n

    The answer is yes but not sure how to justify.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint:

    As \sum_{n=0}^{+\infty}c_n3^n converges, the radius of convergence of the series \sum_{n=0}^{+\infty}c_nx^n is \geq 3
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    Hint:

    As \sum_{n=0}^{+\infty}c_n3^n converges, the radius of convergence of the series \sum_{n=0}^{+\infty}c_nx^n is \geq 3
    I was trying to figure out the radius of convergence. This is what i have.

    we know that |z-a| < R means that the series converges absolutely for some number R.
    Here i have a as (-3) and z=x=0 when it converges.
    so |-3| < R => |3| < R

    now |3| can be + or - 3...so I guess is R > -3 or 3?
    I dunno if im doing this right.
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