Kummer's Test

Given a series of positive terms u_i and a sequence of finite positive constants a_i, let


1. If rho>0, the series converges.

2. If rho<0 and the series sum_(n=1)^(infty)1/a_n diverges, the series diverges.

3. If rho=0, the series may converge or diverge.

The test is a general case of Bertrand's test, the root test, Gauss's test, and Raabe's test. With a_n=n and a_(n+1)=n+1, the test becomes Raabe's test.

See also

Convergence Tests, Raabe's Test

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Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 285-286, 1985.Jingcheng, T. "Kummer's Test Gives Characterizations for Convergence or Divergence of All Series." Amer. Math. Monthly 101, 450-452, 1994.Samelson, H. "More on Kummer's Test." Amer. Math. Monthly 102, 817-818, 1995.

Referenced on Wolfram|Alpha

Kummer's Test

Cite this as:

Weisstein, Eric W. "Kummer's Test." From MathWorld--A Wolfram Web Resource.

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