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Math Help - prove this

  1. #1
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    prove this

    if An>=0 and summation(An) converge, then summation ((An)^2) also converge.please...
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nzm88 View Post
    if An>=0 and summation(An) converge, then summation ((An)^2) also converge.please...
    This is a corollary of a more comprehensive theorem. If \sum_{n\in\mathbb{N}}a_n converges absolutely then \sum_{n\in\mathbb{N}}a_n^2 converges as well.

    To see this merely note that since a_n\to0 there exists some N\in\mathbb{N} such that N\leqslant n\implies |a_n|<1\implies a_n^2<|a_n| and so \sum_{j=N}^{\infty}a_n^2<\sum_{j=N}^{\infty}|a_n| and the conclusion follows.
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    MHF Contributor Bruno J.'s Avatar
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    Well from some point on, say for n>n_0, we will have |A_n|<1. But then A_n^2<|A_n|; so just compare the tails of the two series.
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    MHF Contributor Bruno J.'s Avatar
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    Great minds think alike
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Great minds think alike
    Agreed. We'll probably be in a think-tank someday together. We can then go to bars and try to impress women with our bridling intellect and manish good looks.
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    MHF Contributor Bruno J.'s Avatar
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    It'd be an honour to wing-man you!
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