The Ratio Test
The next tool in our convergence test arsenal is the ratio test. We get the idea from the convergence of geometric series. We mentioned before that geometric series are as common as eating hotdogs. In the case of the ratio test, we want to know if we can safely open our box full of grilled chili-cheese hotdogs.
If Σ an
is a geometric series, we know that the common ratio of the series is
We also know that, if |r| < 1, the geometric series converges.
Now suppose Σ an isn't geometric. Since this series isn't geometric the ratio between successive terms isn't constant, but we're not going to let that stop us. We're going to pretend the series is geometric and look at the ratio between terms anyway. It's like pretending a grilled chili-cheese hotdog is just a hotdog. Pass the wetnaps.
More precisely, we're going to look at the limit of the ratios as n goes to ∞. The ratio test says that, if the ratio of terms
converges to something with absolute value less than 1, the series converges.
Put another way, if the grilled chili-cheese dog isn't too messy, we can eat it without it getting everywhere. It's worth opening the Pandora's box and having a bite.
To show this, we'll stick the absolute value signs inside the limit and look at
In symbols, start with a series
The ratio test says:
- If L < 1 the series converges.
- If L > 1 the series diverges (this includes the case where the limit doesn't exist).
- If L = 1 we can't tell what's going on. In this case we need to find another test to use.
All we're doing, really, is pretending that we're looking at a geometric series.
Before practicing the ratio test, you should be very comfortable with the following ideas.
- To divide a fraction by another fraction we multiply the first fraction by the reciprocal of the second. So
We'll use this a lot because fractions within fractions are icky.
- The factorial expression n! means to multiply together all the numbers from n down to 1 . It's often useful to rewrite factorials. Here are some ways we might want to rewrite them:
(n + 1)! = (n + 1)(n)(n – 1) ×...2 × 1 = (n + 1) n!
(2n + 2)! = (2n + 2)(2n + 1)(2n)(2n – 1) ×...2 × 1 = (2n + 2)(2n + 1)(2n)!
- When we multiply expressions with the same base, we add the exponents . So
2n + 1 = 2n × 21 = 2n × 2
When is the ratio test actually useful? Anytime we encounter a mess of chili and cheese for sure, which corresponds to nasty fractions in terms of math. In particular, we used it for series with fractions where both the numerator and the denominator contained the variable. The ratio test didn't work on the series
where the terms only had the variable in the denominator. The ratio test also didn't work on the series
While there is an n in the numerator, all it does is flip-flop the sign of the term, so this n doesn't really count.
Observation: The ratio test works well on interesting fractions, since we're able to cancel out pieces of the terms to get simpler expressions. These are the ones that easily run out of control, like Velveeta cheese.
Here are the formulas for the terms of the different series on which we were able to use the ratio test:
There are a lot of factorials and exponents in these terms.
Observation: The ratio test works well when the fractions contain factorials and exponents, because we can cancel these out to get simpler expressions. These are the fractions that have so much chili and cheese on them that we need a bowl to catch our drippings as we eat.