convergence of an improper integral

[dts5044]

for what values of the real number x is the improper integral [integral from 1 to infinity] t^(x-1)e^(-t)dt convergent?
my solution was:

1<= t < infinity

suppose x < 1

and define f(t) = t^(x-1)e^(-t) = e^(-t)/t^(1-x) , g(t) = e^(-t)

now x < 1, so x-1 < 0, which implies that 1 - x > 0

and since we are on the interval 1 <= t < infinity, 1/t^(1-x) <= 1

so f(t) <= g(t) for t >= 1, and by the comparison theorem if g(t) converges then so does f(t)

[integral from 1 to infinity] g(t) = [integral from 1 to infinity] e^(-t)dt = 1/e

so g(t), converges for all x < 1, and by the comparison theorem so does f(t)

I used a similar proof using L'hospital's rule t prove the case for x >= 1

but my question is: is this right (for x < 1)??

because say I had chosen g(t) = 1/t^(1-x)

g(t) >= f(t) for all t >= 1 because f(t) = e^(-t)/t^(1-x) and e^(-t) = 1/e^t < 1 for t >=1

so using the comparison theorem with this g(t), if g(t) converges then so does f(t)

[integral from 1 to infinity] g(t) = [integral from 1 to infinity] 1/t^(1-x) *dt

which converges if 1-x > 1 ==> x < 0,

and diverges if 1-x <= 1 ==> 0 <= x

so I'm getting two different answers!! what am I doing wrong!?!?

 

[galactus]

\(\displaystyle \int_{1}^{\infty}t^{x-1}e^{-t}dt\)

You do realize this is the Gamma function for the case t>1?. Except for the small difference of the lower limit being 1 instead of 0.

Here is a way. See what you think. It also uses the comparison test. It appears to be along the lines of your method.

I had this written down from sometime back because I had this back in the day as well.

Let's use the fact that \(\displaystyle \lim_{t\to {\infty}}t^{x-1}e^{\frac{-t}{2}}=0\)

This holds \(\displaystyle \forall \;\ x\).

So, there is a number we can call p so \(\displaystyle t^{x-1}e^{\frac{-t}{2}}<1\) for t>p.

Therefore, \(\displaystyle t^{x-1}e^{-t}=t^{x-1}e^{\frac{-t}{2}}e^{\frac{-t}{2}}<e^{\frac{-t}{2}}\) for t>p.

So, \(\displaystyle \int_{p}^{\infty}t^{x-1}e^{-t}dt\) converges by the comparison test since \(\displaystyle \int_{p}^{\infty}e^{\frac{-t}{2}}dt\)

converges.

 

[dts5044]

okay, I understand this!



but can you tell why my method contradicts itself? I'm okay with your answer I'd just like to see where I went wrong with my method.