
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.

Because $\displaystyle a_n > 0\,\& \,\left( {n \cdot a_n } \right) \to q > 0$ we have $\displaystyle \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \frac{q}
{2} < n \cdot a_n } \right]$.
Do you see that $\displaystyle a_n > \frac{q}{{2n}}$?
This means that $\displaystyle \sum {a_n > \frac{q}{2}\sum {\frac{1}{n}} } $