integraalitestiä

Integraalitestiä, also known as the integral test, is a mathematical tool used in calculus to determine the convergence or divergence of an infinite series. The test applies to series of positive terms. Specifically, if f(x) is a continuous, positive, and decreasing function on the interval [n, infinity) and a_n = f(n) for all integers n, then the infinite series sum from n=1 to infinity of a_n converges if and only if the improper integral from 1 to infinity of f(x) dx converges.

The core idea behind the integral test is to compare the sum of the series to the area under the curve of the corresponding function. For a decreasing function, the area of rectangles under the curve (representing the terms of the series) can be bounded by the area under the curve itself. If the area under the curve (the integral) is finite, then the sum of the areas of the rectangles (the series) must also be finite, implying convergence. Conversely, if the area under the curve is infinite, the sum of the rectangle areas must also be infinite, indicating divergence.

It is crucial that the function f(x) meets all the conditions of the integral test: continuity, positivity, and decreasing nature on the specified interval. If any of these conditions are not met, the integral test cannot be reliably applied to determine the convergence of the series. The starting point of the series and the integral can be shifted, but the convergence or divergence remains the same. For instance, starting the series and integral from n=N to infinity still yields the same conclusion.