show that 
integral(x^p*e^(-kx)) converges for k>0 and p>0

Hi fmunshi!!!

To solve this problem I will use two criterion of convergence, one for
improper integrals and one for series.

- Integral Test:
Consider a decreasing function f:[1,oo)-->[0,oo). 
The improper integral int[f(x), from x=1 to oo] is convergent if and
only if series sum(f(n)) is convergent.
 Note that it may happen that f(x) is not decreasing on the entire
interval [1,oo) but only on some subinterval [A,oo)(where A > 1). The
above conclusion is still valid.
See the proof of this at the SOS Math website: 
"Improper Integrals and Series: The Integral Test"
http://www.sosmath.com/calculus/improper/series/series.html


Now we must check that the function f(x) = x^p*e^(-k.x) (k>0 and p>0)
satisfies the conditions of the integral test.
We need to find if on some interval [A,oo)(where A > 1), f(x) is a
decreasing function, so we need to find some interval [A,oo)(where A >
1) where f´(x) is negative:
f´(x) = p.x^(p-1).e^(-k.x) + x^p.(-k).e^(-k.x) = 
      = e^(-k.x).x^(p-1).(p - k.x)

since p>0, k>0 and we are considering x>0 then f´(x) < 0 if and only
if
(p - k.x) < 0 or the equivalent p < k.x or the equivalent x > p/k .
If A = 1 + p/k we have that f(x) satisfies the condition for the
integral test.

Now we need to prove that sum(f(n)) converges.
For do that we will use the D´alambert Ratio Test:
If S = lim[A(n+1)/An] when n tends to +oo and An > 0 for all n, then
the series sum(An) converges if S < 1, diverges if S > 1 and we do not
have a definite conclusion if S = 1.
For a proof of this criterion see at SOS Math website:
"The Root and Ratio Tests"
http://www.sosmath.com/calculus/series/rootratio/rootratio.html

f(n+1) = (n+1)^p * e^(-k.(n+1))
f(n) = n^p * e^(-k.n)

f(n+1)/f(n) = ((n+1)/n)^p * e^(-k.n - k + k.n) = e^(-k).(1 + 1/n)^p

then 
lim[f(n+1)/f(n)] = lim[e^(-k).(1 + 1/n)^p] = e^(-k). lim[(1 + 1/n)^p]
=
                 = e^(-k) = 1/e^k < 1 (because 1 > e then 1^k > e^k).

Then the series sum(f(n)) converges and for the Integral test the
improper integral int[x^p*e^(-k.x)] (k>0 and p>0) converges.


I hope this helps you, but if you find something unclear and/or need a
clarification, please post a request for a clarification before rate
this answer.

Best regards.
livioflores-ga