Summation of convergent series Sum the series
$$\sum_{n=0}^{\infty}\left(\frac{2}{2n+1} - \frac{1}{n+1}\right)$$
Is there some general method for summing such series?
Hint:

\begin{equation}
 \sum_{n=0}^{\infty}\left(\frac{2}{2n+1} - \frac{1}{n+1}\right)
=
2 \sum_{n=0}^{\infty}\left(\frac{1}{2n+1} - \frac{1}{2n+2}\right)
=
- 2 \sum_{n=1}^{\infty}(-1)^n \frac{1}{n}
\end{equation}

So this is the alternating harmonic series, rest is up to you
We know $\frac12 \log \frac {1+x}{1-x}=x+\frac{x^3}3+\frac{ x^5}5+...$.

$\log ( {1-x})=-x-\frac{x^2}2-\frac{x^3}3-..$.

So,$∑_{n=0}^∞(\frac2{2n+1}−\frac 1{n+1})=\log (1+x)$ at x=1 hence $\log 2$