Could someone help me with the following sum?
$$\sum_1^\infty \frac{1}{2n(2n+1)}$$
I begin as
$$\sum \frac{1}{2n}-\frac{1}{2n+1}=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+...$$
Now, $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...= \ln 2$$
So, $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+...=1-\ln 2$$
Is this correct?
