Real Analysis - Series

**ajj86** · November 8th 2008, 03:37 PM

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.

**Plato** · November 8th 2008, 04:08 PM

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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