Expii

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.