CalcRef

Sequences & Series · Convergence Tests

Integral Test

f(n)=an,  f positive, continuous, decreasing:∑an and ∫1∞f(x) dx both converge or both divergef(n) = a_n,\; f \text{ positive, continuous, decreasing}: \sum a_n \text{ and } \int_1^{\infty} f(x)\, dx \text{ both converge or both diverge}

If f is positive, continuous, and decreasing for x ≥ N, then the series and the improper integral either both converge or both diverge.

Worked examples

Does Σ 1/(n² + 1) converge?
  1. f(x) = 1/(x²+1) is positive, continuous, and decreasing for x ≥ 1
  2. ∫₁^∞ 1/(x²+1) dx = [arctan x]₁^∞ = π/2 - π/4 = π/4 (converges)

Answer: Converges by the integral test.

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