Kummer's Test

Given a Series of Positive terms $u_i$ and a sequence of finite Positive constants $a_i$, let

$$\rho \equiv \lim_{n\to \infty} \left({a_n {u_n\over u_{n+1}}-a_{n+1}}\right).$$

1. If $\rho > 0$, the series converges.

2. If $\rho < 0$, 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

References

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.