The Integral TestConsider the series Σ1/n². How does the area of the rectangles plotted below compare with the sum of the series? Click the checkbox to show the function f(x) = 1/x². Notice that the sum of the areas of the boxes is the sum of the series - but it is also a left Riemann sum for the integral of f(x) from 1 to ∞ (with one box leftover). If you drag the slider over, you can also see that the areas of the boxes make a right Riemann sum for the integral of f(x) from 1 to ∞ (without any leftover). So if the integral makes sense, so does the series! Jason McCullough, Created with GeoGebra |