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The Ratio Test
Calculus 2 · Chapter 7: Convergence Tests for SeriesSection 7.5 · lesson 36 of 5410 min · free
← Absolute and Conditional ConvergenceNext lesson: The Root Test →
In this lesson
Comparing each term to the one before it.
- Compute L = \lim|a_{n+1}/a_{n}| and read off the verdict
- Handle factorials and exponentials with confidence
- Recognise when L=1 makes the test useless
- Route powers-of-n series to a different test
Transcript
Everything said in the video, section by section (1,418 words). Read along, or search it.
MAT-265 - Calculus 2 · Section 7.5
The Ratio Test
The comparison tests measure a series against a benchmark. The Integral Test measures it against an integral. The Ratio Test does something different, and often simpler. It compares each term to the one immediately before it. And if that ratio settles down to a number less than one, the series is behaving like a convergent geometric series — so it converges. This turns out to be the tool of choice for series involving factorials and exponentials. Which is exactly where the earlier tests become awkward.
The Test
Three Outcomes
Here's the theorem. Let the series have nonzero terms for all sufficiently large n. And suppose L is the limit of the absolute value of a n plus one over a n. Suppose that limit exists as a finite number, or is infinite. Then there are three cases. Case one. If L is less than one, the series converges absolutely. And therefore converges. Case two. If L is greater than one, or L is infinite, the series diverges. Case three. If L equals one, the test yields no information at all. The series may converge or may diverge, and a different test must be used. Two details in that statement deserve attention. First, the absolute value bars are not optional. They allow the test to be applied to series with negative or alternating terms. And they're the reason the conclusion in case one is absolute convergence, rather than plain convergence. Second, the conclusion in case two is stronger than it may appear. When L exceeds one, the terms do not approach zero. So the series fails the Divergence Test outright.
Why
It Is a Geometric Comparison in Disguise
Let's see why those three outcomes arise, because the reasoning reveals what the test really is. Suppose L is less than one. Choose any number r strictly between L and one. So L is less than r, which is less than one. Since the ratios converge to L, there's an index N beyond which every ratio stays below r. Now multiply through. The absolute value of a N plus one is less than r times absolute a N. Apply the same inequality again: absolute a N plus two is less than r squared times absolute a N. Continuing that pattern gives absolute a N plus k less than r to the k, times absolute a N. For every positive integer k. So the terms from index N onward are bounded by the terms of a geometric series with ratio r. And that geometric series converges, because r is between zero and one. By the Comparison Test, the tail converges absolutely. And adding the finitely many terms before index N cannot disturb that. So the Ratio Test is really a geometric comparison in disguise. Now suppose L is greater than one. Then beyond some index the ratios exceed one, meaning each term is larger in magnitude than the one before. The magnitudes are increasing from that point on. So they cannot approach zero. And the Divergence Test forces the series to diverge. Finally, L equals one is genuinely uninformative. We'll see exactly why in a moment.
Factorials
Where the Ratio Test Shines
Factorials are where the Ratio Test really earns its place. Because a factorial collapses beautifully inside a ratio. n plus one factorial equals n plus one, times n factorial. So n factorial over n plus one factorial reduces to one over n plus one. No other test handles factorials so cleanly. First example. Does the sum of two to the n over n factorial converge? Form the ratio. Two to the n plus one over two to the n gives two. And n factorial over n plus one factorial gives one over n plus one. So the ratio is two over n plus one. As n goes to infinity, that goes to zero. L equals zero, which is less than one. The series converges absolutely. Second example, with alternating signs. The sum of minus one to the n, times n squared over n factorial. Here's the key move. Take absolute values immediately. The alternating factor has absolute value one in both numerator and denominator. So it simply disappears. What's left is n plus one squared, over n plus one times n squared. Which simplifies to n plus one, over n squared. Divide top and bottom by n squared. The limit is zero. L equals zero, less than one. So the series converges absolutely. And notice — no separate Alternating Series Test was required. Because the Ratio Test works with the absolute ratio, it resolves the alternating series and its absolute-value counterpart in a single computation.
Exponentials
The Base Survives, the Exponent Vanishes
Exponential expressions behave inside a ratio almost as cleanly as factorials. Because b to the n plus one, over b to the n, equals b. For any nonzero base. The base survives. The variable exponent vanishes. And when a series mixes an exponential with a polynomial, something useful happens. The polynomial contributes a ratio that approaches one. So the exponential alone determines L. Let's see it. Does the sum of n over three to the n converge? Form the ratio. You get n plus one over three to the n plus one, times three to the n over n. Separate the polynomial and exponential pieces. The polynomial part is n plus one over n. The exponential part is three to the n over three to the n plus one, which is one third. Take the limit. n plus one over n is one plus one over n, which approaches one. So L is one times one third. One third. Since one third is less than one, the series converges absolutely. And notice that the polynomial factor n was, in the end, completely irrelevant to the conclusion. That's a general pattern worth remembering. Any rational function of n contributes a ratio approaching one.
Diagnostics
Structural Cues That Predict the Outcome
That last observation leads to a practical diagnostic. The table shows which structural features predict which outcome. A factorial in the denominator gives L equals zero. Converges absolutely. A factorial in the numerator gives L infinite. Diverges. A smaller base over a larger base gives L less than one. Converges. A larger base over a smaller base gives L greater than one. Diverges. And then the two shaded rows. Both are series built from powers of n alone. The harmonic series, sum of one over n. And the sum of one over n squared. Both give L equal to one. Yet the harmonic series diverges, and the second series converges. Identical L values, opposite outcomes. Which is exactly why case three must be stated as inconclusive. The ratio simply doesn't carry enough information to separate them. So the practical lesson is diagnostic. Whenever a n is built from powers of n alone — that is, any rational function of n — the ratio will approach one, and the Ratio Test will waste your effort. Route such series to the p-Series Test, the Comparison Test, the Limit Comparison Test, or the Integral Test instead.
Key Takeaways
What L = 1 Actually Tells You
Here's a scenario worth thinking through carefully. A student applies the Ratio Test to the sum of one over n cubed, and correctly computes L equals one. The student then writes: since L equals one, the series diverges. A classmate objects that the series clearly converges — it's a p-series with p equal to three, which is greater than one. Both students computed the same limit, and reached opposite conclusions. So where exactly did the first student's reasoning break down? And what does the value L equals one actually communicate about a series? In framing your answer, consider this. If the Ratio Test could distinguish convergence from divergence at L equals one, what would that imply about the harmonic series and the sum of one over n squared? Both produce identical ratio limits. But opposite behaviors. Five takeaways. Compute L, the limit of the absolute ratio. L less than one gives absolute convergence. L greater than one, or infinite, gives divergence. L equal to one gives nothing. The absolute value bars are essential. They extend the test to alternating and irregular sign patterns, and settle both the series and its magnitudes in one computation. The test is a geometric comparison in disguise. L less than one traps the tail beneath a convergent geometric series. Factorials and exponentials are its natural targets, because both collapse inside the ratio. And a series built only from powers of n always gives L equals one. Don't use the Ratio Test on it. Next up is section seven point six — the Root Test. Which takes an nth root instead of a ratio.
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