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Math Help - Real Analysis - Series

  1. #1
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    Real Analysis - Series

    I posted this one over a week ago, but was hoping that someone might have some input on it.

    a. Show that if an > 0, and lim(n*an) = q with q not equal to zero, then the series Σan diverges.

    b. Assume an > 0 and lim(n^2 * an) exists. Show that Σan converges.
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  2. #2
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    Because a_n  > 0\,\& \,\left( {n \cdot a_n } \right) \to q > 0 we have \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \frac{q}<br />
{2} < n \cdot a_n } \right].
    Do you see that a_n  > \frac{q}{{2n}}?
    This means that \sum {a_n  > \frac{q}{2}\sum {\frac{1}{n}} }
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