Series Convergence Tests: Cheat Sheet and Strategy

You stare at the series and your pencil hangs. Every test you half-remember blurs into the same formula. This reference puts the full set of convergence tests in front of you as a single printable guide, then gives you a repeatable decision method that works for any series you will see in Calculus II or AP Calculus BC.

Series convergence tests are mathematical tools that tell you whether an infinite sum settles to a finite number or grows without bound. The honest version: most series cannot be summed exactly; these tests only tell you if a sum exists, not what it is. What newcomers get wrong most often is that passing one test does not guarantee the series is convergent, the test must be applied correctly, and some series require multiple tests.

Below is the complete cheat sheet for the eight tests you need. Each row gives you the test name, when to use it, the conditions it requires, and what conclusion it can deliver. Print it on one page.

Geometric Series Test

Use when the terms form a constant ratio r between consecutive terms: sum of a * r^n. Converges if |r| < 1. Diverges if |r| ≥ 1. The sum when convergent is a/(1 - r), but note: if the index starts at n=1 instead of n=0, the sum is ar/(1-r).

p-Series Test

Use for series of the form sum 1/n^p. Converges if p > 1. Diverges if p ≤ 1. The harmonic series (p=1) diverges. The Basel problem series (p=2) converges to π²/6.

Divergence Test (nth-Term Test)

Use first on every series. If the limit of a_n as n approaches infinity is not zero, the series diverges. If the limit is zero, the test tells you nothing, you must use another test. Never claim convergence from a zero limit.

Integral Test

Use for positive, decreasing terms where f(n) = a_n and f(x) is continuous. Compare the series to the improper integral of f(x) from 1 to infinity. The series converges if and only if the integral converges. This test does not give the sum; it only gives convergence status.

Comparison Test

Use when you can compare a_n to a known convergent or divergent series b_n. Requires 0 ≤ a_n ≤ b_n (if b_n converges) or a_n ≥ b_n (if b_n diverges). The direction of the inequality matters: reversing it gives no information.

Limit Comparison Test

Use when direct comparison is messy. Compute the limit of a_n / b_n. If the limit is a positive finite number, both series behave the same way (both converge or both diverge). Works when the inequality in the Comparison Test is hard to prove.

Alternating Series Test

Use for series with alternating signs: sum (-1)^n * b_n, where b_n is positive. Converges if b_n decreases monotonically to zero. The decrease must be eventually monotonic, and the limit must be exactly zero. The error after N terms is at most the first omitted term b_{N+1}.

Ratio Test

Use for series with factorials (n!) or exponentials (2^n, e^n). Compute L = limit |a_{n+1} / a_n|. If L < 1, the series converges absolutely. If L > 1, it diverges. If L = 1, the test is inconclusive, you need another test. The Ratio Test is decisive for factorials and exponentials but fails for p-series and most rational functions.

Root Test

Use for series with nth powers like n^n or (something)^n. Compute L = limit (|a_n|)^(1/n). Same rule as Ratio Test: L < 1 converges, L > 1 diverges, L = 1 inconclusive. The Root Test is often easier than the Ratio Test when terms have nth powers.

When you face an unfamiliar series, follow this order. It matches the strategy from Stewart, Calculus, section 11.7 and Paul's Online Math Notes, Strategy for Series.

Step 1: Apply the Divergence Test. Compute the limit of a_n. If it is not zero, the series diverges and you stop. If it is zero, proceed. This saves time more than any other rule.

Step 2: Identify the form. Is it geometric? p-series? Alternating? Factorials or exponentials? Nth powers? Match the form to the test that is built for that form.

Step 3: Apply the matched test. If the test gives a decisive yes or no, record the conclusion. If it is inconclusive (L=1 in Ratio or Root Test, for example), move to Step 4.

Step 4: Reach for the general-purpose tests. Use the Comparison Test, Limit Comparison Test, or Integral Test. These work for rational functions, logarithms, and mixed forms the specialized tests cannot handle.

Step 5: Handle endpoints for power series. If you are working with a power series sum c_n (x-a)^n, apply the Ratio or Root Test to find the radius of convergence R. Then test the endpoints x = a ± R separately using one of the earlier tests.

This is the section that answers the question directly. You have the series in front of you. Here is what to do by form.

Series with Factorials

Factorials are the signature of the Ratio Test. Example: sum n! / 2^n. Compute the limit of a_{n+1}/a_n. The factorial will cancel to something like 1/(n+1) times a constant, which goes to 0. The Ratio Test will be decisive almost every time.

Series with nth Powers

For terms like (3n+1)^n or n^n, use the Root Test. The nth root undoes the power, leaving the base. Example: sum ( (2n+1)/(3n+2) )^n. The Root Test gives L = limit (2n+1)/(3n+2) = 2/3 < 1, so the series converges.

Rational Functions (Polynomials in n)

For series like sum (n^2 + 1) / (n^3 + 2n), use the Limit Comparison Test with a p-series. Compare to 1/n (p=1, diverges) because the leading terms n^2/n^3 = 1/n. The limit of the ratio will be 1, a positive finite number, so the series diverges with the harmonic series. The Comparison Test also works here, but the Limit Comparison Test avoids the inequality headache.

Series with Logs

Logs grow slower than any positive power of n. For sum 1/(n ln n), use the Integral Test because the function f(x) = 1/(x ln x) integrates to ln(ln x), which diverges. For sum (ln n)/n^2, compare to 1/n^{3/2} and use the Limit Comparison Test, the log does not change the limit enough to affect convergence.

Alternating Series

If the series has (-1)^n or (-1)^{n+1}, use the Alternating Series Test directly. Check that the absolute value of the terms decreases (eventually) to zero. If it does, the series converges conditionally. Then test for absolute convergence by applying the Ratio Test or Comparison Test to the absolute value of the terms.

Two examples that show the decision process end to end.

Example 1: sum_{n=1}^{infinity} (n! * 2^n) / (3n)^n

Step 1: Divergence Test. The limit of the term is zero (factorial grows but the denominator grows faster with n^n). Test passes, need more work.

Step 2: Identify the form. Has factorial and n^n. Ratio Test and Root Test are both candidates. Try Ratio Test: a_{n+1}/a_n = ((n+1)! * 2^{n+1})/(3(n+1))^{n+1} divided by (n! * 2^n)/(3n)^n.The limit of the last factor is e^{-1}. Overall limit = 2/3 * e^{-1} ≈ 0.245 < 1. Converges.

Conclusion: Series converges by the Ratio Test.

Example 2: sum_{n=1}^{infinity} (2n^2 + 3) / (4n^3 + 5n + 1)

Step 1: Divergence Test. Limit is 0 (degree 2 over degree 3). Need more work.

Step 2: Identify the form. Rational function. Not geometric, not p-series. Use Limit Comparison Test with b_n = 1/n. Compute limit of (2n^2+3)/(4n^3+5n+1) divided by 1/n = (2n^3 + 3n)/(4n^3 + 5n + 1) → 2/4 = 1/2. Positive finite number.

Conclusion: Series diverges by the Limit Comparison Test with the harmonic series.

These are the errors that cost exam points and waste time.

Misusing the Ratio Test on a p-series. The Ratio Test on sum 1/n^p gives L = 1. The test is inconclusive, yet many students conclude convergence or divergence anyway. You must switch to the p-series test or the Integral Test.

Forgetting the Divergence Test conditions. The Divergence Test only proves divergence when the limit is not zero. If the limit is zero, you have not proven anything. Students who claim convergence from a zero limit are wrong.

Confusing the Comparison Test direction. To prove convergence, you need a_n ≤ b_n where b_n converges. To prove divergence, you need a_n ≥ b_n where b_n diverges. Reversing the inequality tells you nothing.

Using the Alternating Series Test without checking monotonic decrease. The terms must eventually be decreasing, not just approach zero. Example: 1, 1/2, 1, 1/4, 1, 1/6, ... goes to zero but does not decrease monotonically. The test fails.

Assuming absolute convergence from conditional convergence. The alternating harmonic series sum (-1)^n/n converges conditionally. Its absolute value series sum 1/n diverges. The Ratio and Root Tests check absolute convergence. If they show divergence, the series may still converge conditionally, always check absolute convergence first using the absolute value of the terms.