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In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. An early form of the test of convergence was developed in India by Madhava in the 14th century, and by his followers at the Kerala School. In Europe, it was later developed by Maclaurin and Cauchy and is sometimes known as the Maclaurin–Cauchy test.
## Statement of the test

Consider an integer N and a non-negative monotone decreasing function f defined on the unbounded interval [N, ∞). Then the series## Proof

The proof basically uses the comparison test, comparing the term f(n) with the integral of f over the intervals [n − 1, n] and [n, n + 1], respectively.## Applications

The harmonic series
# -frac1{varepsilon x^varepsilon}biggr|_1^M

frac1varepsilonBigl(1-frac1{M^varepsilon}Bigr)
lefrac1varepsilon
quadtext{for all }Mge1.
## Borderline between divergence and convergence

The above examples involving the harmonic series raise the question, whether there are monotone sequences such that f(n) decreases to 0 faster than 1/n but slower than 1/n^{1+ε} in the sense that
_{k} denotes the k-fold composition of the natural logarithm defined recursively by
_{k} denotes the smallest natural number such that the k-fold composition is well-defined and ln_{k} N_{k} ≥ 1, i.e.
# ln_{k+1}(x)bigr|_{N_k}^infty

infty.
To see the convergence of the second series, note that by the power rule, the chain rule and the above result
## References

- $sum\_\{n=N\}^infty\; f(n)$

converges if and only if the integral

- $int\_N^infty\; f(x),dx$

is finite. In particular, if the integral diverges, then the series diverges as well.

Since f is a monotone decreasing function, we know that

- $$

- $$

- $$

- $$

- $$

- $$

- $$

- $$

- $$

Using the integral test for convergence, one can show (see below) that, for every natural number k, the series

- $$

- $$

- $$

- $$

To see the divergence of the first series using the integral test, note that by repeated application of the chain rule

- $$

- $$

- $$

- $$

- Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.3) ISBN 0-486-60153-6
- Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 4.43) ISBN 0-521-58807-3

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Last updated on Thursday August 07, 2008 at 09:54:25 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday August 07, 2008 at 09:54:25 PDT (GMT -0700)

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