Probability- finite n-th moment

[Roni1985]

Homework Statement

Suppose the random variable X has finite exponential moment. Show by comparison to the Taylor series for EXP[x] that X has finite n^th moment (E|X|ⁿ<inf) for all positive integers n

Homework Equations

e^x=$$\sum$$($$\frac{x^n}{(n!)}$$, n,0,inf)

The Attempt at a Solution

we know that E(e^t*x) < inf

I can use the Taylor expansion but I end up with

E(e^t*x) < $$\sum$$($$\frac{x^n}{(n!)}$$

don't know how to pick up from here...

would appreciate any help.

Thanks.

 

[Roni1985]

sorry for the bump, it's really important for me :\ have an exam in two days and I've been trying to solve it since 12 pm.

Thanks.