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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^'=lim_(n->infty)(a_n(u_n)/(u_(n+1))-a_(n+1))
(1)

with a_n=n, giving

rho^'=lim_(n->infty)[n(u_n)/(u_(n+1))-(n+1)]
(2)
=lim_(n->infty)[n((u_n)/(u_(n+1))-1)-1].
(3)

Defining

rho=rho^'+1=lim_(n->infty)[n((u_n)/(u_(n+1))-1)]
(4)

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

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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.

Referenced on Wolfram|Alpha

Raabe's Test

Cite this as:

Weisstein, Eric W. "Raabe's Test." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RaabesTest.html

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