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Two series

Posted: Fri Sep 21, 2012 1:56 pm
by marco
Hello:

I have these two series which I can't handle:

\sum_{k=0}^\inf \frac{k^2}{k!} which according to the GRE exam is 2e. I can't see why.

The other one is:
\sum_{k=0}^\inf \frac{k}{4^(k+1)}. what is its value?

Could someone help me out?
Thanks in advance.

Re: Two series

Posted: Fri Sep 21, 2012 2:40 pm
by blitzer6266
For the first

$$\sum_{k=0}^\infty \frac{k^2}{k!} = \sum_{k=1}^\infty \frac{k}{(k-1)!} =
\sum_{k=0}^\infty \frac{k+1}{(k)!} =
e + \sum_{k=1}^\infty \frac{1}{(k-1)!}$$

Re: Two series

Posted: Fri Sep 21, 2012 2:43 pm
by blitzer6266
for the second use

$$\sum_{k=0}^\infty r^k = \frac{1}{1-r} \Longrightarrow \sum_{k=0}^\infty k r^{k-1} = \frac{1}{(1-r)^2}$$
by taking a derivative. This works for any r in the open unit interval