I'd like to turn this sum:
\begin{align}\sum_{n=0}^{\infty} \frac{x^{n+1}}{3^{n+1}(n+1)} \end{align}
into an integral $\displaystyle \int_{a}^{b} g(x) \space dx$.
There seems to be many methods to either change or approximate sums as integrals. So I've become confused which approach would work.
In Is it possible to write a sum as an integral to solve it? robjohn used $\int_0^\infty e^{-nt}\,\mathrm{d}t=\frac1n$ which looks similar to a Laplace Transforms.
I can't see how he gets rid of the n's so I'm not able to apply it here otherwise it seems promising. But looking elsewhere there are also approximations methods such as: Turning infinite sum into integral which even more obscure at least to me.
How do I convert this sum to an integral?
Best Answer
Well, you could write $$\frac{1}{n+1} = \int_0^1 t^n\; dt$$ so (for $|x| < 3$) your sum becomes $$ \eqalign{\sum_{n=0}^\infty &\left(\frac{x}{3}\right)^{n+1} \int_0^1 t^n\; dt\cr = & \frac{x}{3} \int_0^1 \sum_{n=0}^\infty \left(\frac{xt}{3}\right)^n \; dt\cr = & \frac{x}{3} \int_0^1 \frac{dt}{1-xt/3}\cr = & \ln\left(\frac{3}{3-x}\right)\cr }$$