## A Match Made in Math-Heaven

If series are peanut butter, then integrals are jelly. You could put one or the other on a couple slices of bread and have a satisfying sandwich. You could also put them both together and have a tantalizing treat in your lunch box.

The two go together so nicely for a reason. Although they are very different, they have some nice properties that are very similar. It's no coincidence. Remember that an integral is defined in terms of a limit that the left hand sum (LHS) and the right hand sum (RHS) are the same. An infinite number of intervals is usually used in this limit, so these sums look just like infinite series.

There *is* one difference. To use these properties, we have to know already that the series converge. After all, peanut butter doesn't taste like jelly. They just go well together.

Assuming the series

both converge, then

If

converges and *c* is a constant, then

## Practice:

Find the sum of the series.

You may assume

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).

Find the sum of the series.

You may assume

Answer

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

Find the sum of the series.

You may assume

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:

Find the sum of the series.

You may assume

Answer

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

Find the sum of the series.

You may assume

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.

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.