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Math Help - Divergent and Convergent summs\series

  1. #1
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    Divergent and Convergent summs\series

    1) \displaystyle\sum_{n=1}^{\infty}{a_n} is divergent.

    Proove that \displaystyle\sum_{n=1}^{\infty}\frac{a_n}{1+n*a_n  } is also divergent.


    2) a_n \geq a_{n+1} \textgreater 0 and \displaystyle\sum_{n=1}^{\infty}{a_n} is convergent.

    Proove that: \lim\limits_{n \rightarrow \infty}{n*a_n}=0

    I am a third year student at Computer Science, therefore I have some strong basics. Any help, direction where to look is appreciated. Thx

    LE:
    3) <br /> <br />
\sum_{i=0}^\infty \arctan\frac{1}{n^2+n+1}=?<br />
    Last edited by Yeti; October 25th 2009 at 07:18 AM.
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  2. #2
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    2) a_n \geq a_{n+1} \textgreater 0 and \displaystyle\sum_{n=1}^{\infty}{a_n} is convergent.

    Proove that: \lim\limits_{n \rightarrow \infty}{n*a_n}=0
    Let  S_n = \sum_{i=1}^n a_n, S = lim_{n\to\infty}S_n. Pick any epsilon > 0. Then there exists a K such that for any k >= K, (a_(K+1) + ... + a_m) <= S - S_k < epsilon/2 since S_k converges to S and since the partial sums are increasing. Also, since a_n converges to zero, there must exist a J > K such that for all j >= J, (epsilon/2)*min{a_1, a_2, ..., a_K}/S_K >= a_j. Therefore since the sequence is decreasing, for all sufficiently large j, n\cdot a_j = \underbrace{a_j + \ldots + a_j}_{n\;\rm times} \leq (\frac{\epsilon a_1}{2S_K} + \ldots + \frac{\epsilon a_K}{2S_K}) + (a_{K+1} + \ldots + a_j) = \frac{\epsilon S_K}{2S_K} + (a_{K+1} + \ldots + a_j) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon . Since the sequence is positive but eventually remains smaller than any positive number, it must converge to zero.
    Last edited by rn443; October 25th 2009 at 06:35 AM.
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