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We've already mentioned that series and integrals are much alike. The integral is the series' brother from another mother. Before we put on our x-ray glasses and take a closer look at convergence and divergence of series, we need to review and discuss two things: basic integral definitions, convergence, and divergence; and visualizing sequences and series.

Since we've already discussed everything we need to know about integrals, we've listed the important concepts here:

- weighted area, including left- and right-hand sums

- improper integrals with an infinite limit of integration

- what it means for such integrals to converge or diverge

- comparison of improper integrals

- the p-test for integral convergence

If you feel unsure about any of these, be sure to go review before we dive headfirst into the new stuff.

We already know we can visualize a sequence on a 2-D graph. We make the horizontal axis show which term we're on and the vertical axis show the value of that term.

Because they're sort of flat, we're going to use grilled cheese sandwiches to visualize series in 2-D.

First, we have a difficult question for you to answer; it's sure to confuse you for hours. What happens to the value of a number if you multiply it by 1?

The value of the number doesn't change. We sincerely hope you got that question right.

What is the area of a grilled cheese sandwich with height *a _{n}* and width 1?

Answer.

The area of the sandwich looks like this:

Its area is

(height)(width) = *a _{n}*(1) =

### When Limits of Summation Don't Matter

Now that we know we can visualize series using rectangularly-shaped sandwiches, we can use these visualizations to discuss convergence and divergence of the series. We already mentioned that if we only care whether or not a series converges (that is, we don't care what it converges

*to*)it's ok to write

Σ

*a*_{i}without limits of summation.

In other words, moving the starting index up and down doesn't change whether the sum of the series exists or not, although it will affect the sum we get if the series converges. This means that we don't care what our first sandwich looks like. We only care what the later sandwiches look like. That's a relief. We burned the first couple of grilled cheeses in the last series.

For this discussion, we're going to cheat by only thinking about series with positive terms and American-type grilled cheeses. This makes the pictures come out better: all the rectangles will be on top of the horizontal axis, and the only way for a series to diverge is for its partial sums to approach ∞.

We are going to assert a few claims using positive terms so we can prove them easily, but they are valid if we use positive and/or negative terms. Showing that is a bit more involved than you probably care about.

Make

*M*<*N*. After all, they are ordered that way in the alphabet. Also, we should remember that any finite series converges. In particular,is finite, so its sandwiches must make up a finite food area.

**Claim:**If converges then so does .### Proof:

If converges then its rectangles make up a finite region.

If we chop off part of the finite region, for example, our burned sandwiches from the beginning, what's left is still finite:

This means

converges also.

**Claim:**If converges then so does .### Proof:

If converges, then its rectangles make up a finite region.

If we stick on another finite region, we have a bigger (but still finite) region. More food for everyone, but still finite amounts.

Putting the first two claims together tells us that,

*if a series converges, we can change the starting index to anything we like and the resulting series will still converge.*For example, if we can show thatconverges, then we know

converges too.

**Claim:**If diverges then so does .### Proof:

If diverges then its rectangles must cover an infinite area.

If we chop off a finite area from the beginning, our burnt grilled cheese sandwiches, what's left must still be infinite.

This means

diverges.

**Claim:**If diverges then so does .### Proof:

If diverges then its rectangles must cover an infinite area. Sticking on a little piece of finite area still leaves us with an infinite area. This time, we don't get more food to share. No one cares, because there's plenty of grilled cheese to go around anyway.

This means

diverges too.

Putting those two claims together,

*if a series diverges, we can change the starting index to anything we like and the resulting series will still diverge.*For example, if

diverges then so does