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Gauss's Test

If u_n>0 and given B(n) a bounded function of n as n->infty, express the ratio of successive terms as

|(u_n)/(u_(n+1))|=1+h/n+(B(n))/(n^r)

for r>1. The series converges for h>1 and diverges for h<=1 (Arfken 1985, p. 287; Courant and John 1999, p. 567).

Equivalently, with the same conditions as above, given

|(u_(n+1))/(u_n)|=1-p/n+(B(n))/(n^r),

the series converges absolutely iff p>1 (Zwillinger 1996, p. 32).

See also

Convergence Tests

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 1. New York: Springer-Verlag, 1999.Wrede, R. C. Schaum's Outline of Advanced Calculus, 2nd ed. New York: McGraw-Hill, p. 268, 2002.Zwillinger, D. (Ed.). "Convergence Tests." §1.3.3 in CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, 1996.

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Gauss's Test

Cite this as:

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

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