- #1

sapiental

- 118

- 0

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 [tex]\int \frac {1}{x^p}dx [/tex]

convergent?

from 1 to infinity [tex]\int \frac {1}{x^p}dx [/tex]

= lim (t -> infinity) [tex]\frac {x^-^p^+^1}{-p+1} [/tex] (from x = 1 to x = t)

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

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) [tex]\frac {t^p^-^1}{p-1} - \frac {1}{p-1}[/tex]

Thanks!

from 1 to infinity [tex]\int \frac {1}{x^p}dx [/tex]

convergent?

from 1 to infinity [tex]\int \frac {1}{x^p}dx [/tex]

= lim (t -> infinity) [tex]\frac {x^-^p^+^1}{-p+1} [/tex] (from x = 1 to x = t)

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

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) [tex]\frac {t^p^-^1}{p-1} - \frac {1}{p-1}[/tex]

Thanks!

Last edited: