Show two inequalities - (context gamma function converges)

[binbagsss]

Homework Statement

I'm not after another proof.
I've just got a couple of inequalities I don't know how to show when following a given proof in my book.

These are:

Q1) $0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1}$

So this is obvioulsy true, however I think I'm being dumb because surely this is true for all $x$

(the integral over the gamma function is split into $\int ^{\infty} _1$ and $\int^1_0$) and so this inequality is used in the latter, but I don't see why it can't be used in the former, e.g $2^{t-1}/e^{2} \leq 2^{t-1}$ is also true isn't it? and as $x \to \infty$ this is also true, because the LHS $\to 0$..

Q2) that $x^{t-1}e^{-x}\leq x^{-x/2}$, for $x \geq 1$
I am unsure how to show this, or understand why it holds,

and then secondly I need to show that $x^{t-1}e^{-x}\leq x^{-x/2} \iff x^{t-1}\leq e^{x/2}$

I can write $x^{-x/2} = e^{(-x/2) ln (x) }$
I am unsure what to do next or whether this helps

Homework Equations

see above

The Attempt at a Solution

see above

 

[pasmith]

Homework Statement

I'm not after another proof.
I've just got a couple of inequalities I don't know how to show when following a given proof in my book.

These are:

Q1) $0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1}$

So this is obvioulsy true, however I think I'm being dumb because surely this is true for all $x$

(the integral over the gamma function is split into $\int ^{\infty} _1$ and $\int^1_0$) and so this inequality is used in the latter, but I don't see why it can't be used in the former, e.g $2^{t-1}/e^{2} \leq 2^{t-1}$ is also true isn't it? and as $x \to \infty$ this is also true, because the LHS $\to 0$..

$$\int_1^R x^{t-1}\,dx = \begin{cases} \ln R & t = 0, \\<br /> \frac{R^t - 1}{t} & t \neq 0\end{cases}.$$ For $t \geq 0$ these do not converge as $R \to \infty$. You're trying to prove convergence so this bound is of no assistance.

 

[binbagsss]

$$\int_1^R x^{t-1}\,dx = \begin{cases} \ln R & t = 0, \\<br /> \frac{R^t - 1}{t} & t \neq 0\end{cases}.$$ For $t \geq 0$ these do not converge as $R \to \infty$. You're trying to prove convergence so this bound is of no assistance.

thank you, makes sense !
Q2 anyone?

 

[binbagsss]

thank you, makes sense !
Q2 anyone?

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