Ratio Test

Ratio Test with Factorials

PJ Using the Ratio Test to Determine if a Series Converges #3 (Factorials)

The Ratio Test is a tool we can use to indicate convergence of a series. The Ratio Test is particularly useful with factorials (!) and with n-th powers.

Given a series $\sum{a_n}$ , we can calculate

$L=\lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right|$

Then we can draw the following conclusions:

If $L<1$ , the series is absolutely convergent
If $L>1$ , the series is divergent
If $L=1$ , we can’t draw a conclusion from the ratio test, and therefore must use another test in order to determine the convergence of a series

To use the formula, plug the original function into the denominator of the fraction in the formula for $L$ . Substitute $n+1$ into your original function for every appearance of the variable, then plug that into the numerator of the fraction in the formula for $L$ .

Now simplify the fraction as much as possible. If your function includes factorials, try writing out the first several terms of each factorial, and then canceling what you can. For example, write out $n!$ as $n(n-1)(n-2)(n-3)…$ . If you do this for every factorial in your series, you may be able to cancel some terms. Similarly, if you have something in your function like $\frac{3^n}{3^{n+1}}$ , remember that you can divide the term in the numerator into $3^n3^1$ , thereby allowing you to cancel $3^n$ from both the numerator and denominator.

Ratio Test

Ratio Test with Factorials

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