Let . For there is an with and for all .

Since is a bijection, there is an such that all occur among . Hence for , the rearranged partial sum contains every term of , and whatever remains comes from the small tail:

So the rearranged series converges to as well.

For a conditionally convergent series the order of summation matters:

Rearranged as

the series sums to half its original value: two positive terms are traded for one negative term each round, so convergence is preserved but the sum changes.