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 ai
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 that
converges, 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
