Series Exercises

Answer

Since both and converge, we can break up the series:

If  or  diverged, we wouldn't be allowed to break up the series like that (and it wouldn't help us anyway).

Answer

Since  converges and 5 is a constant, we're allowed to pull the constant out of the summation sign:

Answer

Remember that -b_n is the same thing as (-1)b_n. Since  converges, we can pull the (-1) out of the summation sign:

Answer

This is a combination of parts (a) and (c). Since both  and  converge (this is necessary!), we can break up the sum:

Answer

Since Σ c_n diverges, we aren't allowed to break up the sum. However, in this case, we don't have to. The quantity

c_n – c_n

is always zero, so

Adding up zero infinitely many times gives us zero.

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.