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Sequences & Series · Convergence Tests

Divergence Test

lim⁡n→∞an≠0⇒∑an diverges\lim_{n \to \infty} a_n \neq 0 \Rightarrow \sum a_n \text{ diverges}

If the terms do not approach zero, the series diverges. CAUTION: If the limit IS zero, the test is inconclusive (the series may still diverge).

Worked examples

Does Σ n/(2n+1) converge?
  1. lim(n→∞) n/(2n+1) = 1/2 ≠ 0

Answer: Diverges by the divergence test.

Can the divergence test determine if Σ 1/n converges?
  1. lim(n→∞) 1/n = 0. Test is inconclusive.

Answer: Inconclusive (the series actually diverges, but this test cannot show it).

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