Ah right, that was my assumption that it was true for some real number greater than 1. I should have been more explicit.
To prove that \Gamma(p+1) converges for 0\leq p <1 couldn't I just say that \Gamma(1+1) \geq \Gamma(0+1) and since \Gamma(1+1) converges then the gamma function must...
I just learned induction in another thread and I'm curious if it can be used to prove that the gamma function converges for p\geq0. I'm not sure if it can be used in this way. Is this wrong?
Gamma Function is defined as:
\Gamma(p+1)=\int_0^\infty e^{-x}x^p \,dx We're trying to show that this...
So I could just do this
(D-r)^{m-1} (m-1) x^{m-2} e^{rx}=(m-1)(D-r)^{m-1} x^{m-2} e^{rx}
Then I just say
(D-r)^{m-1} x^{m-2} e^{rx}=0
so
(m-1)(D-r)^{m-1} x^{m-2} e^{rx}=0
right?
Is that all I have to show? The original problem used k=0,1,...,m-1. Does this prove it for all the other...
I've been working on this for the past couple of days and I'm just going around in circles.
I'm trying to prove
Q(m)=(D-r)^m x^{m-1} e^{rx}=0
So using induction I prove this for m=1
Q(1)=(D-r)^1 x^0 e^{rx}=D(e^{rx})-re^{rx}=0
Then I assume that the following is true
Q(m-1)=(D-r)^{m-1}...
No I never learned induction so I was trying my best with what I had. And it should have been x^(m-2) in my previous post. For splitting the operand up, I just factored a (D-r) out and made an assumption, which may have been false, that (D-r) is commutative. Plus it just seemed to work out...
I think I was a little too liberal with copy and pasting in the original.
It should have read:
(D-r)(x^ke^{rx})=D(x^ke^{rx})-rx^ke^{rx}=kx^{k-1}e^{rx}
I'm not sure why I can't split up the operand. I tried to get it to work but I couldn't so I started down a different path along my...
Unfortunately we never learned induction and the instructor knows this...don't ask me how we did series without it. :confused:
I did a little reading and from what I gather you prove the case for n=1, then assume the theorem is true for n-1 and doing some algebraic manipulation you can...
How would I prove that (D-r)^m annihilates x^ke^{rx} where k=0,1,...,m-1.
At first I tried to do it as a computation by expanding the whole the thing out but that turned out to be a nightmare. After that I went down a different path that seems promising but I can't seem to figure it out...
1) Show that since a body falling freely obeys the differential equation [tex]h''=-g[/itex], if it falls from an initial height h(0), it lands with a velocity of -\sqrt{2gh(0)}
This problem is from a differential equations class and I solved it two different ways:
Method 1:
\frac {dh}...