ratio test

For a series

inf
---
\ an
/__
n=1

if lim_n->inf |a_n+1/a_n| = L < 1, the series is absolutely convergent.
If lim_n->inf |a_n+1/a_n| = L > 1, the series diverges.
If lim_n->inf |a_n+1/a_n| = L = 1, the ratio test tells us nothing, and we must use another test.

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) = x², 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!/(xx is it convergent?

            (x+1)!
           ------
f(x+1)     (x+1)(x+1)         (x+1)! xx            xx
-----  =  ------------ =  --------------  =  ---------
f(x)          x!            x!(x+1)(x+1)        (x+1)x
           ------
             xx     

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,  Fx is absolutely convergent
If |l| > 1,  Fx 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.

If the basic convergence tests (see the infinite series node for a list of essential tests and tips) do not yield anything and you cannot easily apply the Cauchy condition to determine the convergence of an infinite series, then you might want to give d'Alembert's ratio test a try.

Although "less powerful", it is much easier to apply, when facing certain complex sequences (especially the ones filled with nasty exponents):

Let there be a series Σu_n where (u_n) is a strictly positive sequence (u_{n >} 0 for all n).

If you can find a real k positive and strictly less than 1 ( k ∈ [0,1[ ) and an integer N such as:

 u_n+1
------ < k for all n ≥ N
  u_n

Then Σu_n converges.

On the other hand:

If you can find a real r strictly greater than 1 ( r > 1 ) and an integer N such as:

 u_n+1
------ > r for all n ≥ N
  u_n

Then Σu_n diverges.

The "intuitive" way of putting this test (please bear with the imprecision inherent to an attempt to explain rather abstract mathematic concepts into layman's terms) is that, if after a certain rank (N), the sequence's terms keep getting a bit "closer" together with each increment of n (hence the importance of the strict relations), the series will converge. While if each term in the sequence is a bit "farther" from (or equal to) the previous one, the sequence will never "stabilize" and will therefore diverge.

The proof to this test relies on the Cauchy condition.

What's Next?
If this test does not yield any conclusive result on the convergence of your series, you might want to look at Raabe's test or Cauchy Condition (and make sure you have previously tried all the trivial conditions listed under the infinite series node).

Extra Bonus Mathematics Trivia: I bet you did not know that Mr. Jean Le Rond d'Alembert also happened to be co-author (along with Denis Diderot) of L'Encyclopedie, an 18th century attempt at collecting an exhausive and detailed catalog of all forms of human knowledge that can arguably be deemed the first serious precursor to E2. Well now you know.