<s>[INST] 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?
 [/INST] Solution 1:
Prove by induction that 
$$1-\frac 12 +\frac 13 - \frac 14 + \cdots + \frac{1}{2n-1}-\frac{1}{2n}=\frac{1}{n+1}+...+\frac{1}{2n}$$
Then use the fact that $\frac{1}{n+1}+...+\frac{1}{2n}=\frac{1}{n}\sum_{k=1}^n \frac{1}{1+\frac{k}{n}}$ is a Riemann sum.
Solution 2
$$\begin{align}
u_n &=1-\frac 12 +\frac 13 - \frac 14 + \cdots + \frac{1}{2n-1}-\frac{1}{2n}\\&=1+\frac 12 +\frac 13 + \frac 14 + \cdots + \frac{1}{2n-1}+\frac{1}{2n}-2 (\frac{1}{2}+...+\frac{1}{2n})\\&=\left(1+\frac 12 +\frac 13 + \frac 14 + \cdots + \frac{1}{2n-1}+\frac{1}{2n} -\ln (2n)\right)\\&\qquad-\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}{n}-\ln(n)\right)+\ln(2)
\end{align}$$
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