Mathematics
Integral Test
Quick fact
The integral test shows that the harmonic series 1 + 1/2 + 1/3 + ... diverges because its related integral diverges, while the p-series with p=2 converges.
Why this is interesting
You know how some sums, like 1 + 1/2 + 1/3 + ..., seem to keep growing forever, but others, like 1 + 1/4 + 1/9 + ..., approach a finite number? The integral test gives a clever way to tell which is which using the idea of area under a curve.
Read the full explanation
Understanding Integral Test
Imagine a decreasing function f(x) that is always positive, like 1/x. Draw rectangles with width 1 under the curve from x=1 to infinity. The sum of the areas of these rectangles approximates the integral (area under the curve). But the integral is easier to compute. The integral test says: if the improper integral of f(x) from 1 to infinity converges (has a finite area), then the sum of the series ∑ f(n) converges. If the integral diverges, so does the series.
A deeper explanation
The test works because for a decreasing continuous function f, the integral from 1 to N of f(x) dx is bounded between the sum from 1 to N-1 and the sum from 2 to N. As N → ∞, the series and integral either both converge or both diverge. This is proven by comparing rectangle sums (Riemann sums) with the integral. The test is crucial because it connects discrete sums (series) to continuous calculus, allowing us to evaluate series like ∑ 1/n^p by computing ∫ x^{-p} dx. It's the foundation for other tests like the p-series test.