# list of convergent series

I wanted to know if there is an online reference I can use to find out known results about convergent series. I could not find this one, for example, on wikipedia

$\sum_{k=1}^{+\infty} \left(\frac{1}{2}\right)^k k^2$

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No, but there is a list of tests for convergence, which would be much more useful for someone taking a test on this topic... – Antonio Vargas May 12 '12 at 23:01
Are you trying to work out whether it converges, or are you trying to work out what it converges to? – Gerry Myerson May 13 '12 at 12:17

There exists an $N$ such that for all $k > N$, $k^2 \le (3/2)^k$. This is just because $$\lim_{k \to \infty} \frac{(3/2)^k}{k^2} = \lim_{k \to \infty} \left( \frac{\sqrt{3/2}^k}{k} \right)^2 = \infty, \quad \Longrightarrow \quad \exists N \mbox{ s.t.} \frac{(3/2)^k}{k^2} \ge 1 \mbox{ for all k<N}.$$ Therefore, $$\sum_{k=N+1}^{\infty} \frac{k^2}{2^k} \le \sum_{k=N+1}^{\infty} \frac{(3/2)^k}{2^k} = \sum_{k=N+1}^{\infty} \left( \frac 34 \right)^k$$ which converges.
Without saying more about $N$, your sums should start with $k=N+1$, no? – Matthew Conroy May 13 '12 at 3:47