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


\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


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.

© 1996-9 Eric W. Weisstein