Show that if $\sum b_n$ is a rearrangement of a series $\sum a_n$ , and $a_n$ diverges to $\infty$, then $\sum b_n = \infty$

Question

I was presented with the following problem;

Show that if $\sum b_n$ is a rearrangement of a series $\sum a_n$ , and $a_n$ diverges to $\infty$, then $\sum b_n = \infty$.

How would one solve this? It seems intuitively true, but how could I show it?

Answer 1

I suppose you mean that $a_n\to +\infty$. Under this assumption, we can proceed as follows.

Let $M>0$ be an arbitrary number. Then there is some $N>0$ such that $a_n>1$ for any $n>N$. Since $\{b_n\}$ is a rearrangement of $\{a_n\}$, then there is some $N'>0$ such that $a_1,\ldots,a_N$ are contained in $b_1,\ldots,b_{N'}$.

Set $B=\sum_{k=1}^{N'}b_k$. Then for any $T>|B|+M+N'$, we have $$\sum_{k=1}^{T}b_k=\sum_{k=1}^{N'}b_k+\sum_{k=N'+1}^{T}b_k\ge B+(T-N')>M.$$

Therefore, $\sum b_n\to+\infty$.

If we only know $\sum a_n=+\infty$, we can say nothing about $\sum b_n$. See Riemann rearrangement theorem.