# Sum of two divergent series

1. Sep 12, 2015

Consider the two divergent series:
$$\sum_{n=k}^{\infty} a_n$$
$$\sum_{n=k}^{\infty} b_n$$
Is it possible for $\sum_{n=k}^{\infty} (a_n \pm b_n)$ to converge?

2. Sep 12, 2015

### Orodruin

Staff Emeritus
Yes, consider the case $a_n = b_n$. Then the sum of $a_n - b_n$ is zero.

3. Sep 12, 2015

Thanks!
But in that case does it make sense to say that $\sum_{n=k}^{\infty} (a_n \pm b_n) = \sum_{n=k}^{\infty} a_n \pm \sum_{n=k}^{\infty} b_n$?

4. Sep 12, 2015

### Orodruin

Staff Emeritus
No, it generally make sense only if the series converge.

5. Sep 12, 2015

### PeroK

You really ought to be able to find examples yourself to resolve this question.

6. Sep 13, 2015

### Ssnow

You can construct others, for example $a_ {n}=\frac{1}{n^{2}}+b_{n}$ so $\sum_{n=1}^{\infty}a_{n}-b_{n}=\frac{\pi^{2}}{6}$...