Series Convergence Tests: Step-by-Step Guide
A series sum of a_n converges if its partial sums approach a finite limit; otherwise it diverges. There is no single test that handles every series, so you select a test based on the form of the terms: nth-term test, geometric, p-series, comparison, ratio, root, alternating, or integral test.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Apply the nth-term test first. If lim a_n is not 0, the series diverges immediately. If the limit is 0, this test is inconclusive and you must try another test.\[\lim_{n\to\infty} a_n \neq 0 \;\Rightarrow\; \sum a_n \text{ diverges}\]
- Recognize standard forms. Geometric series sum r^n converges iff |r| < 1. p-series sum 1/n^p converges iff p > 1. These cover many common cases.\[\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges} \iff p > 1\]
- Try the ratio or root test for factorials/exponentials. For terms with n!, k^n, or n^n, compute L = lim |a_(n+1)/a_n| (ratio) or L = lim |a_n|^(1/n) (root). If L < 1 converges, L > 1 diverges, L = 1 inconclusive.\[L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right|\]
- Use comparison or limit comparison for rational forms. Compare to a known p-series or geometric series. Limit comparison: if lim a_n/b_n = L > 0 finite, both converge or both diverge.\[\lim_{n\to\infty} \frac{a_n}{b_n} = L,\; 0 < L < \infty\]
- Use the alternating series test if signs alternate. If a_n is positive, decreasing, and a_n -> 0, then sum (-1)^n a_n converges.\[\sum (-1)^n a_n \text{ converges if } a_n \downarrow 0\]
Worked example
Determine whether the series sum from n=1 to infinity of n / 2^n converges or diverges.
Step 1: Check the nth-term test: lim n/2^n. By L'Hopital or growth rates, exponential beats polynomial, so the limit is 0. The test is inconclusive.
Step 2: The terms involve 2^n, so try the ratio test. Let a_n = n/2^n.
Step 3: Compute a_(n+1)/a_n = ((n+1)/2^(n+1)) * (2^n/n) = (n+1)/(2n).
Step 4: Take the limit: lim (n+1)/(2n) = 1/2.
Step 5: Since L = 1/2 < 1, the ratio test concludes the series converges.
Verification: This is a standard convergent series; in fact sum n/2^n from n=1 to infinity equals 2.
Tips and common mistakes
- Always start with the nth-term test - it's the fastest way to detect divergence.
- If a_n -> 0, the series may still diverge (e.g., harmonic series), so don't conclude convergence from that alone.
- Use the ratio test when factorials or exponentials appear in a_n.
- Use the root test when a_n has form (something)^n.
- For rational expressions in n, compare to a p-series with p = (degree of denominator) - (degree of numerator).
- The alternating series test only confirms convergence; check absolute convergence separately.
- Integral test requires a_n = f(n) where f is positive, continuous, and decreasing.
- When in doubt, try multiple tests; some series satisfy several tests.