Step By Step Calculus » 17.3 - Ratio and Root Tests

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Synopsis
Herein, we discuss the Ratio Test and Root Test for checking convergence of nonnegative series. Both tests compare the series of interest with a geometric series.
Ratio Test: For a non-negative series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i suppose that \lim_{n\to \infty} \frac{a_{n+1}}{a_n}=\lambda\in[0,\infty]\lim_{n\to \infty} \frac{a_{n+1}}{a_n}=\lambda\in[0,\infty]
Then, the series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i
(i) converges if \lambda<1\lambda<1,
(ii) diverges if \lambda>1\lambda>1 or \lambda=\infty\lambda=\infty,
(iii) may converge or diverge if \lambda=1\lambda=1.
In the last case, the test is inconclusive.
Root Test: For a nonnegative series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i suppose that \lim_{n\to \infty} \sqrt[n]{a_n}=\lambda\in[0,\infty]\lim_{n\to \infty} \sqrt[n]{a_n}=\lambda\in[0,\infty]
Then, the series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i
(i) converges if \lambda <1\lambda <1,
(ii) diverges if \lambda >1\lambda >1 or \lambda=\infty\lambda=\infty,
(iii) may converge or diverge if \lambda=1\lambda=1.
In the last case, the test is inconclusive.