Integral test for convergence

The integral criterion (also integral criterion for comparison ) is a mathematical convergence criterion for infinite series. The range is regarded as the area under a staircase function which is estimated by the area under a curve. With an estimate upwards allows the convergence to prove down the divergence. The area under the curve is calculated by the integral.

Formulation

It is a monotonically decreasing function, which is defined on the interval with an integer number and assumes only positive values ​​. Then the series converges, when the integral exists, that is, when it assumes a finite value.

More: Be monotonically decreasing, then applies

If one of the two, ie existence of the integral or convergence of the series, and thus the latter is the case, the estimates are

Example

You want to check the given function with whether its associated sum

Converges. is monotonically decreasing in the interval and the integral criterion is obtained finally:

The integral is finite and thus the series has to be convergent.

Illustration

The integral criterion is already accessible through intuition: Just the last line is similar to a popular justification of the concept of Riemann integral using upper and lower sums.

Because by assumption falls monotonically yes, on every interval ( by an integer ) is the largest and the smallest function value in this interval. Because the interval has a width of 1, the area under always less than or equal to and greater than or equal. Now, if the integral converges or the series, as well as the other one expression must converge.

Or: The series converges, so approaches from infinitely close to the limit. For the integral of this means that the surface is not larger but also approximates a (surface ) value. Had the surface to infinity no limit, never a value for the integral could be made fixed and thus assume the integral no finite value, which is in contradiction to the above definition.