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 an and/or bn?
Suppose converges. What happens?
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
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 an and bn are.