How to interpret the sum of two series?

Question

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?

Answer 1

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.

Let us recall that a series is the limit of the sequence of the partial sums.

What you are actually doing in your example is the rearrangement of the terms of the series, which does not necessarily give you the same limit unless the series is absolutely convergent. By rearranging a conditionally convergent series you could actually get anything. This is the famous Riemann series theorem.

I hope this helps.