Which Convergence Test Should I Use?

∞ X 1. Is the series a geometric series? I.e. arn. n=0 Then: Converges for |r| < 1 and diverges otherwise. ∞ X 1 2. Is the series a p-series? I.e. . np n=0 Then: Converges for p > 1 and diverges otherwise. 3. Check the nth Term Divergence Test. Then: Diverges if lim an 6= 0. No information, otherwise. n→∞

4. Is the series a positive series? I.e. an ≥ 0 for all n. Then: Check the following tests, in order. (a) The Direct Comparison Test: Can you find a larger series that converges or a smaller series which diverges? This is usually done by dropping terms from the numerator or denominator. 1 1 1 1 I.e. √ < , but √ > . n2 + n n2 n2 − n n2 Then: Larger series converging implies convergence, and smaller series diverging implies diver- gence. (b) The Limit Comparison Test: Can you find a comparison with a known convergent or divergent series, but the inequality is the wrong way for the Direct Comparison Test? Consider the “dom- 1 1 inant” term in the series. I.e. √ is dominated by n2, so compare with . n2 − n n2 an X X Then: Consider lim = L. If L > 0, an and bn both converge or both diverge. If n→∞ bn X X X L = 0, convergence of bn implies convergence of an. If L = ∞, convergence of an X implies convergence of bn. (c) The Ratio Test: Does the series contain terms with factorials (i.e n!) or exponentials (i.e. 2n)?

an+1 X X Then: Consider lim = L. If L < 1, an converges absolutely. If L > 1, an n→∞ an diverges. If L = 1, no information. (d) The Root Test: Does the series contain a term of the form f(n)g(n) (i.e. n2n)? 1/n pn X Then: Consider lim |an| = lim |an| = L. If L < 1, an converges absolutely. If L > 1, n→∞ n→∞ X an diverges. If L = 1, no information.

(e) The Integral Test: If all other tests for positive series are unusable, consider an = f(n). Is f(n) decreasing and continuous for all n after some M? X Z ∞ Then: an converges if and only if f(x) dx converges. M 5. Does the series have any negative terms? Then: Consider the following in order. ∞ X n−1 (a) Alternating Series Test: Does the series have the form (−1) bn, where {bn} is positive, n=1 decreasing, and converges to 0? ∞ X n−1 Then: (−1) bn converges. n=1 X (b) Does the series converge absolutely? I.e. |an| converges. X Then: an also converges.