Can we find $ \lim_{n \to \infty } n\left ( \frac{1}{n} - \frac{1}{n+1} + \frac{1}{n+2} - \frac{1}{n+3} + ... \right ) $? I have got one method,
If we consider $ a_{n} = \int_{0}^{1} \frac{nx^{n-1}}{1+x} \ dx $
Then, $ \lim_{n \to \infty } n\left ( \frac{1}{n} - \frac{1}{n+1} + \frac{1}{n+2} - \frac{1}{n+3} + ... \right ) = \lim_{n \to \infty }a_{n} = \frac{1}{2} $
But can anyone attack this problem in a different & more standard way?

Like @sansae
$$L=  \lim_{n \to \infty } n\left ( \frac{1}{n} - \frac{1}{n+1} + \frac{1}{n+2} - \frac{1}{n+3} + ... \right )$$
$$\implies L=\lim_{n \to \infty}n\left(\frac{1}{n(n+1)}+\frac{1}{(n+2)(n+3)}+\frac{1}{(n+4)(n+5)}+...+\frac{1}{(n+k)(n+k+1)}+...+\right)$$
But conver the limit to integral as
$$\implies L= \lim_{n \to \infty}\frac{1}{n} \sum_{k=0}^{n} \frac{1}{(1+k/n)(1+(k+1)/n)}=
\int_{0}^{1} (1+x)^{-2} dx=\frac{1}{2}.$$
