CalcRef

Sequences & Series · Convergence Tests

Root Test

L=lim⁡n→∞∣an∣n:L<1⇒conv.,  L>1⇒div.,  L=1⇒inconclusiveL = \lim_{n \to \infty} \sqrt[n]{|a_n|}: \quad L < 1 \Rightarrow \text{conv.},\; L > 1 \Rightarrow \text{div.},\; L = 1 \Rightarrow \text{inconclusive}

The root test: compute the limit of the nth root of |aₙ|. Useful when terms involve nth powers.

Worked examples

Does Σ (n/(2n+1))ⁿ converge?
  1. |aₙ|^(1/n) = n/(2n+1)
  2. lim n/(2n+1) = 1/2 < 1

Answer: Converges by the root test (L = 1/2).

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Ratio Test

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