Topic 10.8 Ratio Test for Convergence: use the limit of the ratio of consecutive terms to decide absolute convergence or divergence, especially for factorials and exponentials (BC).

What this topic is asking

The College Board (Topic 10.8, BC only) gives the ratio test, the most powerful general test in the unit and the standard tool for series with factorials or exponentials. It examines the limit of the ratio of consecutive terms; if the terms shrink fast enough, the series converges absolutely.

The test

Why factorials and exponentials are its specialty

Factorials and $n$-th powers collapse in the ratio because consecutive terms differ by a single factor. For factorials, $\frac{(n+1)!}{n!} = n+1$, so a tower of multiplications reduces to one term. For powers, $\frac{r^{n+1}}{r^n} = r$, a constant. This is why the ratio test handles $\sum\frac{2^n}{n!}$ effortlessly: the ratio $\frac{2}{n+1}\to 0$, immediately giving convergence. By contrast, applying the ratio test to a $p$-series gives $\frac{a_{n+1}}{a_n} = \left(\frac{n}{n+1}\right)^p\to 1$, the inconclusive case, which is why you classify $p$-series by their own rule rather than the ratio test. Matching the test to the term's structure, ratio for factorials and powers, $p$-series rule for powers of $n$ in the denominator, is the strategic point.

A worked convergence

Setting up the ratio without slips

The mechanical heart of the test is writing $a_{n+1}$ correctly: replace every $n$ in $a_n$ by $n+1$. So if $a_n = \frac{3^n}{n!}$, then $a_{n+1} = \frac{3^{n+1}}{(n+1)!}$, and forming the quotient lets the $3^n$ and $n!$ cancel against their successors. Two reliable simplifications: $\frac{(n+1)!}{n!} = n+1$ and $\frac{r^{n+1}}{r^n} = r$. Flip the division by the denominator into multiplication by its reciprocal to keep the algebra clean. A common error is mishandling the factorial, for instance writing $\frac{(n+1)!}{n!} = n!$ instead of $n+1$; expanding $(n+1)! = (n+1)\cdot n!$ makes the cancellation obvious.

When the ratio test is inconclusive

When $L = 1$, the ratio test gives no information, and this happens precisely for the borderline series the test is not built for, chiefly $p$-series and other algebraic (non-exponential, non-factorial) terms. In those cases you fall back on the $p$-series rule, comparison, or the integral test. Recognizing the inconclusive case early saves effort: if the term is a ratio of polynomials in $n$, the ratio test will return $1$, so go straight to a comparison or $p$-series argument instead. Reserve the ratio test for terms whose growth is exponential or factorial, where it gives a clean limit other than $1$, and it will rarely disappoint.