Integral: For What Values of p is Convergent?

[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!

 

[radou]

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

 

[sapiental]

Oh, I'm sorry the intital inegral is

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

thanks for catching that mistake :)