Integral Test - Expii

We can compare series to not just (simpler) series, but also (simpler) integrals. The most common version of the integral test says that, if \(f\) is a non-negative decreasing function such that \(f(n) = a_n\) for all positive integers \(n\), then the series \(\sum_{n=1}^{\infty} a_n := \lim_{t\to\infty} \sum_{n=1}^{t}a_n\) converges if and only if the integral \(\int_1^\infty f(x)\,dx := \lim_{t\to\infty} \int_1^t f(x)\,dx\) (an improper integral with endpoint \(\infty\)) converges.