Probability- finite n-th moment

[roniy]

Suppose the random variable X has finite exponential moment. Show by comparison to the Taylor series for EXP[x] that X has finite nth moment (E|X|[sup:uxb9t7o7]n[/sup:uxb9t7o7]<inf) for all positive integers n

e[sup:uxb9t7o7]x[/sup:uxb9t7o7]=SUM (x[sup:uxb9t7o7]n[/sup:uxb9t7o7]/(n!)) for n=0,1,2,.... inf

I know that
e[sup:uxb9t7o7]x[/sup:uxb9t7o7] < SUM(|x[sup:uxb9t7o7]n[/sup:uxb9t7o7]|/(n!)

same holds for
E[e[sup:uxb9t7o7]x[/sup:uxb9t7o7]] < E[SUM(|x[sup:uxb9t7o7]n[/sup:uxb9t7o7]|/(n!)]

but don't know what to do next.
Would appreciate any help.

Thanks.

 

[mmm4444bot]

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