📂Analysis

Definition of Improper Integrals

Definition¹

Assume that the function $f$ is integrable over every interval $[a,b]$ with fixed $a$ and $b>a$. If the following limit exists, then it is defined as the improper integral of $f$.

$$ \int _{a}^{\infty} f(x) dx = \lim \limits_{b \to \infty} \int _{a}^{b} f(x)dx $$

In this case, if the integration on the left-hand side converges, and if replacing $f$ with $\left| f \right|$ the limit still exists, it is said to converge absolutely.

Explanation

Regarding improper integrals, there is a theorem called the Integral Test.

Theorem

Assume that the function $f$ is $f(x) \ge 0$ and is monotonically decreasing on the interval $[1, \infty)$. Then the following holds.

$$ \int_{1}^{\infty}f(x)dx \text{ converges} \iff \sum\limits_{n=1}^{\infty}f(n) \text{ converges} $$