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
$\sum_{k=1}^{+\infty} \left(\frac{1}{2}\right)^k k^2$
$\endgroup$ 23 Answers
$\begingroup$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.
If you're looking for some library which lists all known patterns of convergence series, you're not trying to understand the mathematics behind those series correctly. The right way to study this (for example, for an exam) is to not only understand the tricks used to show that one series converges/diverges, but also to understand why this was the chosen trick and that it wasn't done purely by luck. Also I suggest trying to apply every elementary tests a few times to know when they work and when they don't.
Hope that helps,
$\endgroup$ 4 $\begingroup$Note that $$ \sum\limits_{1\, \le \;k} {\left( {{1 \over 2}} \right)^{\,k} k^{\,2} } = \sum\limits_{1\, \le \;k} {k^{\,2} x^{\,k} } \quad \left| {\;x = 1/2} \right. $$ and you can find $$ F(x) = \sum\limits_{1\, \le \;k} {k^{\,2} x^{\,k} } $$ by yourself, if you know that $$ \eqalign{ & G(x) = \sum\limits_{0\, \le \;k} {x^{\,k} } = {1 \over {1 - x}}\quad \Rightarrow \cr & \Rightarrow \quad x{d \over {dx}}x{d \over {dx}}G(x) = \sum\limits_{1\, \le \;k} {k^{\,2} x^{\,k} } = {{x\left( {x + 1} \right)} \over {\left( {1 - x} \right)^{\,3} }} \cr} $$
or otherwise through a plenty of resources on-line.
$\endgroup$ $\begingroup$Use the ratio criteria. This is the limit when K tends to infinit, for the term evaluated at K+1 over the term evaluated at K. As the result is 1/2 < 1, then the series converges
$\endgroup$ 1
