Raabe's Test

Given a Series of Positive terms $u_i$ and a Sequence of Positive constants $\{a_i\}$, use Kummer's Test

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

with $a_n = n$, giving

$$\rho' \equiv \lim_{n\to\infty} \left[{n{u_n\over u_{n+1}}-(n+1)}\right]$$

$$= \lim_{n\to\infty} \left[{n\left({{u_n\over u_{n+1}}-1}\right)-1}\right].$$

Defining

$$\rho\equiv \rho'+1=\lim_{n\to\infty} \left[{n\left({{u_n\over u_{n+1}}-1}\right)}\right],$$

then gives Raabe's test:

1. If $\rho > 1$, the Series Converges.

2. If $\rho < 1$, the Series Diverges.

3. If $\rho = 1$, the Series may Converge or Diverge.

See also Convergent Series, Convergence Tests, Divergent Series, Kummer's Test

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 286-287, 1985.

Bromwich, T. J. I'a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 39, 1991.