# Integral: For What Values of p is Convergent?

• sapiental
In summary, the conversation is about finding the values of p for which the integral from 1 to infinity of \frac{1}{x^p}dx converges. The participant provides a solution involving limits and clarifies a mistake in the initial integral.
sapiental
I got lost in an example in my book. Hoping someone could explain it to me.For what values of p is the intergral

from 1 to infinity $$\int \frac {1}{x^p}dx$$

convergent?

from 1 to infinity $$\int \frac {1}{x^p}dx$$

= lim (t -> infinity) $$\frac {x^-^p^+^1}{-p+1}$$ (from x = 1 to x = t)

= lim (t -> infinity) $$\frac {1}{p-1} [\frac {1}{t^p^-^1} - 1]$$

the only thing that confuses me about this is how the t^p-1 ended up in the denominator because after the 2nd sept I get the following:

= lim (t -> infinity) $$\frac {t^p^-^1}{p-1} - \frac {1}{p-1}$$

Thanks!

Last edited:
Isn't $$\int \frac{1}{x^{-p}}dx=\frac{x^{1+p}}{1+p}$$?

Oh, I'm sorry the intital inegral is

$$\int \frac {1}{x^p}dx$$

thanks for catching that mistake :)

## What is the definition of "Integral: For What Values of p is Convergent?"

The term "integral" refers to a mathematical concept that represents the area under a curve in a graph. "For What Values of p is Convergent?" is asking for the values of a variable that would result in the integral being a convergent or finite value.

## Why is it important to know the values of p for which the integral is convergent?

Knowing the values of p for which the integral is convergent is important in understanding the behavior of the function and determining its convergence or divergence. It also helps in solving various mathematical problems involving integrals.

## What is the general formula for finding the values of p for which the integral is convergent?

The general formula for finding the values of p for which the integral is convergent is p > 1. This means that the integral is convergent when the value of p is greater than 1.

## What happens to the integral if the value of p is less than or equal to 1?

If the value of p is less than or equal to 1, the integral becomes divergent or infinite. This means that the area under the curve cannot be calculated or is infinite.

## Are there any exceptions to the general formula for finding the values of p for which the integral is convergent?

Yes, there are some exceptions to the general formula. For example, when dealing with improper integrals, the values of p for convergence may be different. It is important to consider the specific conditions and limitations of the integral in order to accurately determine the values of p for convergence.

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