Let be a convergent series and be a divergent series. Does the series

converge or diverge? Does the answer depend on the particular choice of *a*_{n} and/or *b*_{n}?

Hint

Suppose converges. What happens?

Answer

Following the hint, suppose

converges. Since we know converges also, we can say

The left-hand quantity is the difference of two convergent series, which is some finite number.

The right-hand quantity is

which diverges.

We're claiming a finite number is equal to the sum of a divergent series. This is a problem, because a divergent series can't have a finite sum. We must have started with a bad assumption, so

must not converge after all. It doesn't matter what *a*_{n} and *b*_{n} are.