Question

What do we mean when we say that an improper integral converges? What do we mean when we say that an improper integral diverges?

Short answer

If the limits involved exist, then we say that the improper integral converges to the value determined by those limits, and if the limits are infinite or fail to exist, then we say that the improper integral diverges.

Step-by-step solution

Step 1. Given information

We need to explain that when we say that an improper integral converges and diverges.

Step 2. Definition for Converges and diverges in the improper integral

If the limits involved exist, then we say that the improper integral converges to the value determined by those limits, and if the limits are infinite or fail to exist, then we say that the improper integral diverges.

Step 3. Examples for convergent and divergent.

The example for Convergent improper integral is,

${\int }_1^{\infty }\frac{1}{x^2}dx$

The value of this integral is $1$, which is finite.

The example for divergent improper integral is,

${\int }_1^{\infty }\frac{1}{x}dx$

The value of this integral is $\infty$, which is not define.