The Comparison Test
The last convergence tool we have is the comparison test. If all else fails, we should compare our Pandora's box to another one. If we look at the other one, and we decide the other one is bursting at the seems, we know it's safe to open ours up.
We studied improper integrals a while back, and we learned that, if f ≤ g on the interval (c, ∞), then
Here, when f and g are nonnegative, there's a smaller area under f and than under g. If the big area is finite, the smaller area must be finite too. If the small area is infinite, the bigger area must be infinite. If the other box is smaller and it's too dangerous to open, then ours is too. If is the other box is larger and it is safe to open, so is ours.
Since we can visualize a series as an area , we can use the same intuition to compare series. Suppose we have two series
0 ≤ an ≤ bn
for all n. Then the small area described by the series a is contained in the big area described by the series B.
This tells us two useful things:
- If the area described by B is finite, the smaller area described by a must also be finite.
- If the area described by a is infinite, the bigger area described by B must also be infinite.
Be Careful: As with improper integrals, we have to be careful about which way the comparisons go.
- If the area described by a is finite, that doesn't tell us anything useful. The area described by B could be a larger finite value, or it could be infinite.
- If the area described by B is infinite, that doesn't tell us anything useful. The smaller area described by a could be either finite or infinite.
The tricky part, as with improper integrals, is finding the correct series to compare with. We can't compare gremlins to grilled cheese. We need something similar and easy to tell if the series converge or diverge. We like to use series of the form
whenever possible, since we can easily tell whether such series diverge or converge.
Be Careful: We can't use the comparison test if we can't find something to compare with. For example, we can't use the comparison test on
The only thing we can see to compare the term with is , but the inequality goes the wrong way: