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Math Help - convergence of 1/n as a sequence, but not as a series, why?

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    convergence of 1/n as a sequence, but not as a series, why?

    The sequence, 1/n converges to 0 right, because the limit as n-? is 0.

    So why does the series of 1/n diverge? Doesn't The sequence's convergence imply that at some point, the series will simply be adding 0's to a finite number?
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  2. #2
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    Re: convergence of 1/n as a sequence, but not as a series, why?

    Quote Originally Posted by StudentMCCS View Post
    The sequence, 1/n converges to 0 right, because the limit as n-? is 0.
    So why does the series of 1/n diverge? Doesn't The sequence's convergence imply that at some point, the series will simply be adding 0's to a finite number?
    Define the nth partial sum to be {H_n} = \sum\limits_{k = 1}^n {\frac{1}{k}}.

    Now with induction it is easy to show that \forall N,~H_{2^N}\ge 1+\frac{N}{2}~.

    Thus H_n\to\infty~.
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