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Concept
Integral test for convergence
Summary
In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
Consider an integer N and a function f defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series
converges to a real number if and only if the improper integral
is finite. In particular, if the integral diverges, then the series diverges as well.
If the improper integral is finite, then the proof also gives the lower and upper bounds
for the infinite series.
Note that if the function is increasing, then the function is decreasing and the above theorem applies.
The proof basically uses the comparison test, comparing the term f(n) with the integral of f over the intervals
and , respectively.
The monotonous function is continuous almost everywhere. To show this, let . For every , there exists by the density of a so that . Note that this set contains an open non-empty interval precisely if is discontinuous at . We can uniquely identify as the rational number that has the least index in an enumeration and satisfies the above property. Since is monotone, this defines an injective mapping and thus is countable. It follows that is continuous almost everywhere. This is sufficient for Riemann integrability.
Since f is a monotone decreasing function, we know that
and
Hence, for every integer n ≥ N,
and, for every integer n ≥ N + 1,
By summation over all n from N to some larger integer M, we get from ()
and from ()
Combining these two estimates yields
Letting M tend to infinity, the bounds in () and the result follow.
The harmonic series
diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get
On the other hand, the series
(cf. Riemann zeta function)
converges for every ε > 0, because by the power rule
From () we get the upper estimate
which can be compared with some of the particular values of Riemann zeta function.
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