Root Test

Q: How do you use the Root Test?

Worked example. Does Σ (n/(2n+1))ⁿ converge? |aₙ|^(1/n) = n/(2n+1). lim n/(2n+1) = 1/2 < 1. Answer: Converges by the root test (L = 1/2).

L = \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.

2 worked examples
18 in Sequences & Series

Worked examples

Does Σ (n/(2n+1))ⁿ converge?

|aₙ|^(1/n) = n/(2n+1)
lim n/(2n+1) = 1/2 < 1

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

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Does Σ 2ⁿ/nⁿ converge?

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Common questions

What is the Root Test?

The root test: compute the limit of the nth root of |aₙ|. Useful when terms involve nth powers. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Root Test?

Worked example. Does Σ (n/(2n+1))ⁿ converge? |aₙ|^(1/n) = n/(2n+1). lim n/(2n+1) = 1/2 < 1. Answer: Converges by the root test (L = 1/2).

Practice

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