Question: I'd like to ask if this equality is true: $$\lim\_{N\to \infty} \sum\_{n=N}^{\infty}|a\_n| = lim\_{n\to \infty} |a\_n|$$
It seems to me to be correct, but I do not know how to prove it.

**Edit**: I see that it is not true in every case, but it must be true if the series converges, why is it so?

Answer: 
This is not true, a counter example is
$$
a\_n=\frac1n,\qquad \lim\_{n\to \infty}|a\_n|=0, \qquad \lim\_{N\to \infty} \sum\_{n=N}^{\infty}|a\_n|=\lim\_{N\to \infty} \sum\_{n=N}^{\infty}\frac1n=\infty.
$$
It is true if either 

a) $\displaystyle \sum\_{k=1}^\infty |a\_n|$ converges, by virtue of the fact that both quantities are zero, or 

b) $|a\_n| \to \infty$.