It appears that you did the integration by parts incorrectly.
Let . Then .
So,
, so
Apply the induction assumption to the RHS.
The question is:
Prove (by mathematical induction) that:
I've proved the equality for n=1 as my base step. For my induction step I said that S(1) is true. Suppose that S(n)=S(k). We need to prove that S(k+1) is true. So the LHS becomes:
And I used integration by parts to get the expression
However I'm not sure what the RHS expression becomes for S(k+1). I thought it might be:
But I don't think that's correct. Any help appreciated! Many thanks.
CP
Hi SlipEternal, thanks for your response. I did the same as you for the LHS. I got:
. We know that:
. So for the LHS I end up with:
I then multiplied this out to get what I stated in my first post. My problem is getting the RHS to equal this!
CP
EDIT: I just realised I may have made a mistake in multiplying the last expression out. I'll give it another go and make sure. But my RHS problem remains.
You can't just say "S(1) is true". If n= 1 then S(1) is . Did you do the integral on the left "by parts"?
What is shown "on the right" is .Suppose that S(n)=S(k). We need to prove that S(k+1) is true. So the LHS becomes:
And I used integration by parts to get the expression
However I'm not sure what the RHS expression becomes for S(k+1). I thought it might be:
But I don't think that's correct. Any help appreciated! Many thanks.
CP
Replacing "n" by "k+ 1" that becomes .
That is what you have- although for some reason you have the first two terms in the numerator reversed.