Evaluate $\displaystyle{\lim_{n\to \infty}}(1-\frac 12 +\frac 13 - \frac 14 + \cdots + \frac{1}{2n-1}-\frac{1}{2n}) $ I want to calculate this limit.
$$\lim_{n\to \infty}(1-\frac 12 +\frac 13 - \frac 14 + \cdots + \frac{1}{2n-1}-\frac{1}{2n}) $$
I tried to pair the terms of the sum in order to reduce each other but without any succes. By writing the sum:
$$\frac{1}{1*2} + \frac{1}{3*4}+\frac{1}{5*6}+\cdots+\frac{1}{2n(2n-1)}$$
I have not reached anything useful. Could you help me?

Hint.
$$
x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots=\int_0^x\left(1-t+t^2-t^3-t^4+\cdots\right)\,dt=\int_0^x\frac{dt}{1+t}=\log(1+x)
$$
From this, one can obtain that
$$
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots=\ln 2
$$
