I don't succed to evaluate this sum $\sum_{n=1}^{+\infty}\frac{n}{n^4+4}$ using integration because using fraction method is too hard for me and i have got that is equal $\frac 3 8$ , Is there any way to use integral for Evaluation ?
Asked
Active
Viewed 105 times
1
-
This is a telescoping series...that really seems like the easiest way of summing it. – lulu Feb 26 '18 at 00:34
-
https://math.stackexchange.com/questions/632615/the-number-n4-4-is-never-prime-for-n1 – lab bhattacharjee Feb 26 '18 at 01:30
1 Answers
7
It's telescopic. By Sophie Germain's identity $$n^4+4 = (n^2+2)^2-(2n)^2 = (n^2-2n+2)(n^2+2n+2),$$ so $$ \frac{1}{n^2-2n+2}-\frac{1}{n^2+2n+2} = \frac{1}{f(n)}-\frac{1}{f(n+2)} = \frac{4n}{n^4+4} $$ and $$ \sum_{n\geq 1}\frac{n}{n^4+4}=\frac{1}{4}\sum_{n\geq 1}\left(\frac{1}{f(n)}-\frac{1}{f(n+2)}\right) = \frac{1}{4}\left(\frac{1}{f(1)}+\frac{1}{f(2)}\right)=\frac{3}{8}. $$
Through the inverse Laplace transform:
$$\begin{eqnarray*} \sum_{n\geq 1}\frac{n}{n^4+4}&=&\lim_{\varepsilon\to 0^+}\frac{1}{2}\sum_{n\geq 1}\int_{0}^{+\infty}\sin(s)\sinh(s)e^{-(n+\varepsilon)s}\,ds\\&=&\frac{1}{4}\lim_{\varepsilon\to 0^+}\int_{0}^{+\infty}(1+e^{-s})\sin(s)e^{-\varepsilon s}\,ds\\&=&\frac{1}{4}\left(1+\frac{1}{2}\right)\end{eqnarray*}$$ we reach the same conclusion.
Jack D'Aurizio
- 353,855
-
Where'd you come up with the inverse Laplace transform? It seems to come out of thin air. – Thomas Davis Feb 26 '18 at 01:01
-
@user458276: it is just a standard way (at least in analytic number theory) for converting series into integrals. – Jack D'Aurizio Feb 26 '18 at 01:02