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Integral Test


Let sumu_k be a series with positive terms and let f(x) be the function that results when k is replaced by x in the formula for u_k. If f is decreasing and continuous for x>=1 and

 lim_(x->infty)f(x)=0,
(1)

then

 sum_(k=1)^inftyu_k
(2)

and

 int_t^inftyf(x)dx
(3)

both converge or diverge, where 1<=t<infty. The test is also called the Cauchy integral test or Maclaurin integral test.


See also

Convergence Tests

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 283-284, 1985.Zwillinger, D. (Ed.). "Convergence Tests." §1.3.3 in CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, p. 32, 1996.

Referenced on Wolfram|Alpha

Integral Test

Cite this as:

Weisstein, Eric W. "Integral Test." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntegralTest.html

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