Why the order of a series can't be changed

Question

Why do the infinite series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-... \neq 1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}...$ ?
While when integrating it's possible to change the order of the integration?

Answer 1

Have you heard of the Riemann Rearrangement theorem? You can certainly do this if you know that your series converges absolutely.

Note that your series does not converge absolutely: If you take the absolute value of each term, then you have harmonic series, which is a divergent one. Hence, you cannot do what you wanted to do. :p