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Math Help - Infinite series

  1. #1
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    Infinite series

    Any help for the convergence or not of the infinite series below?

    a. Sum[k^k/k!, {k,1,inf}]
    b. Sum[k!/3^k, {k,1,inf}]

    Thanks a lot!
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  2. #2
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    Quote Originally Posted by jmytil View Post
    a. Sum[k^k/k!, {k,1,inf}]
    Use ratio test, (for suficiently large k), [it diverges]
    The second one also diverges.
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  3. #3
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    Hello, jmytil!

    Darn, I wish LaTeX was back on-line . . .



    . . . .k^k
    a) . .-----
    . .k=1 .k!

    Ratio Test . . .

    a_(n+1) . . . . (k + 1)^{k+1} . . k! . . . . . (k + 1)^{k+1} . . 1 . . . . .(k + 1)^k
    ---------- . = . ------------------ . ------ . = . ------------------ . ----- . = . -----------
    . .a_n . . . . . . . . (k+1)! . . . . . k^k . . . . . . k + 1 . . . . . . k^k . . . . . .k^k


    . . . . . . . .= . [(k + 1)/k]^k . = . [1 + 1/k]^k


    R . = . lim (1 + 1/k)^k . = . e
    . . . . .k→∞


    Since R > 1, the series diverges.



    I'll let someone else explain part (b) . . . I need a nap!
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