The Ratio Test is a way to tell the convergence of the sum of an infinite series. A series is a set of numbers, and infinite means it goes on forever. So, you ask, wouldn't the sum of a set of numbers that go on forever be infinitely large too? For most functions this is true, f_{(x)} = x, for example, goes on forever, as does f_{(x)} = x^{2}, however f_{(x)} = x!/(x^{x}) does not expand infinitely. It has a limit of the irrational number e. The ratio test, however, does not tell you where the function converges, but it does tell you if it does converge at a number.

The ratio test is pretty simple to do. Take f_{x+1} / f_{x}. Confused? I'll explain with an example and explanation.

f_{(x)} = x!/(x^{x} is it convergent?
(x+1)!
------
f_{(x+1)} (x+1)^{(x+1)} (x+1)! x^{x} x^{x}
----- = ------------ = -------------- = ---------
f_{(x)} x! x!(x+1)^{(x+1)} (x+1)^{x}
------
x^{x}

I know I know, y'all are going, what? but let's work through this. The first fraction is the set up. It's f_{x_1} over f_{x}. The next fraction is fleshing out the first fraction. Simply putting in what the function equals. We have a fraction divided by another fraction, so you should remember to invert the denominator of the "big" fraction, and multiply. Do that correctly and you'll end up with fraction 3. In fraction 3 a lot cancels out. This is true for the majority of ratio tests, it depends on the function, of course. x+1! is equal to x+1 times x!, so x! cancels out, leaving x+1, but in the denominator is (x+1)^{(x+1)} which is nothing more than (x+1) times (x+1), x+1 times. This means an (x+1) can be canceled, leaving (x+1)^{x}. Got it?

Now, we're dealing with limits, so in front of all the equations there sould be limit of x as x goes to infinity. This means that fraction four simplifies into a ginormous number over another ginormous number that just a smidge bigger. This simplifies to just under one, or 1^{-}. Now, we're at the decision point in the ratio test. We'll call the limit as x goes to infinity l.

If |l| < 1, F_{x} is absolutely convergent
If |l| > 1, F_{x} is NOT convergent
If |l| = 1, then the test is inconclusive, and another test must be used

In this case, l < 1, so the function is convergent. The ratio test is nice because it is a quick method to figure out convergence. This has several uses in calculus, and appeared first in my Calculus II class. It's nice because it's real simple, set up the fraction, invert the denominator, multiply, cancel, then set the limit to infinity and simplify. It's simple, really ;^)

###### Source:

Calculus Concepts and Contexts, by James Stewart. The second edition.