# How do I determine if the improper integrals converge or not?

## I narrowed the options down to A and C since Bk must converge for k > 1. How do I do the same for Ak?

Jun 14, 2018

The answer is $C$

#### Explanation:

Considering ${A}_{k}$

If the value of $k$ is greater than $1$, you will have something of the form

${\left[\frac{1}{x} ^ k\right]}_{0}^{1}$

This won't work, because $\frac{1}{0}$ is undefined which makes the integral diverge.

This eliminates options A, D and E.

Considering ${B}_{k}$

This is your basic p-series. Just like the above integral yields something close to ${\left[\frac{1}{x} ^ k\right]}_{0}^{1}$, this one will be as follows:

${\lim}_{t \to \infty} {\left[\frac{1}{x} ^ k\right]}_{1}^{t}$

Recall that ${\lim}_{t \to \infty} \frac{1}{t} ^ k = 0$ so this converges (has a finite value).

However, if we're of the form $0 < k < 1$, the numbers will be massive and the integral will diverge.

This means the correct answer is C.

Hopefully this helps!