For what values of r does [tex]\int[/tex](from 0 to infinity) xre-x dx converge?
I assume that the problem refers to r as any real number.
2. The attempt at a solution
I have given this a try but im really not confident that I did it right....
First i used integration by parts to try to discover a pattern:
[tex]\int[/tex]xre-xdx = -xre-x-[tex]\int[/tex]rxr-1(-e-x)dx
I wont write out the whole thing since I cant find all the appropriate summation/product symbols, but carrying out this exact integration by parts an infinite number of times gives:
-e-x(a polynomial in x with an infinite number of terms)
(Note: the polynomial will have a finite # of terms if r happens to be a natural number)
Now I know that the -e-x term will always go to zero as x gets larger.
So I consider the xr, xr-1, xr-2,... terms:
If r[tex]\leq[/tex]0, then those terms will all go to zero, but if r>0 then some of those terms will go to infinity.
From this I concluded that the integral converges iff r[tex]\leq[/tex] 0.
Does this solution make any sense? Thanks in advance to anyone who is able to help me out. I apologize if its not clear; I wrote it much more clearly (and in more detail) on paper but this is my first try at typing math on a computer so i couldnt figure out how to express some things.