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Math Help - Absolute Convergence

  1. #1
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    Absolute Convergence

    I need help determining if these alternating series converge absolutely, converge conditionally or diverge. Give reasons.

    a) the sum (from n=1 to infinity) of (-1)^n [tanh n / n ]
    note tanh n>0 for all n>0


    b) the sum (from n=1 to infinity) of (-1)^n [sinh(n / 8n+1)]
    note: sinh n > 0 for all n > 0

    Any help with these problems is greatly appreciated!

    K
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by kithy View Post
    I need help determining if these alternating series converge absolutely, converge conditionally or diverge. Give reasons.

    a) the sum (from n=1 to infinity) of (-1)^n [tanh n / n ]
    note tanh n>0 for all n>0
    For large n, \tanh(n) goes to 1, so the summand behaves like \frac{(-1)^n}{n}.

    So this is conditionaly convergent, but not absolutely convergent.

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by kithy View Post
    I need help determining if these alternating series converge absolutely, converge conditionally or diverge. Give reasons.

    b) the sum (from n=1 to infinity) of (-1)^n [sinh(n / 8n+1)]
    note: sinh n > 0 for all n > 0
    Here the summand behaves like (-1)^n \sinh(1/8) for large n

    RonL
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