# How to interpret the sum of two series?

I am confused a bit while I am recalling the infinite series. Thomas' Calculus says: Sum of two divergent series can be convergent by giving an example: $\sum1 + \sum-1 = \sum0=0$. We also know that $\sum(-1)^n$ is divergent. However, can not we think the series $\sum(-1)^n=-1+1-1+1\cdots$ equals to $\sum1 + \sum-1$? What distinguishes these series exactly?

• In $\sum 1 + \sum -1$, you can say that number of $1$'s and $-1$'s will be the same which is not in the case of $\sum(-1)^n$ – Isha Tarte Feb 27 '17 at 10:38
• It seems that you are trying to re arrange the terms of $\sum (-1)^n$? – Juniven Feb 27 '17 at 10:44
• @IshaTarte Yes, it seems that their numbers not equal and this leads my question actually. For the latter one, can not we correspond the $-1$'s and $1$'s one-to-one which means their numbers are equal? Why? – faith Feb 27 '17 at 10:57
• @ΘΣΦGenSan Yes this is exactly what I try to. – faith Feb 27 '17 at 10:57
• For your information, rearranging such terms is not allowed. See the answer bellow for a better explanation. – Juniven Feb 27 '17 at 11:08

Suppose that we have two sequences $\{a_n:n\ge1\}$ and $\{b_n:n\ge1\}$ and we want to find the limit of the sum of these two sequences. Then $$\lim_{n\to\infty}(a_n+b_n)=\lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n$$ provided that both of the sequences $\{a_n:n\ge1\}$ and $\{b_n:n\ge1\}$ converge. If this is not the case, the equality might not hold.