Help needed with mathematical induction problem

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

Re: Help needed with mathematical induction problem

It appears that you did the integration by parts incorrectly.

Let . Then .

So,

, so

Apply the induction assumption to the RHS.

Re: Help needed with mathematical induction problem

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.

Re: Help needed with mathematical induction problem

Quote:

Originally Posted by

**CrispyPlanet** 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.

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"?

Quote:

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

What is shown "on the right" is .

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.

Re: Help needed with mathematical induction problem

Quote:

Originally Posted by

**CrispyPlanet** 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

Do you understand what the ... means? If for instance, your expression means . What you multiplied out is only the first four factors.

Re: Help needed with mathematical induction problem

Quote:

Originally Posted by

**SlipEternal** Do you understand what the ... means? If

for instance, your expression means

. What you multiplied out is only the first four factors.

Thanks for that. I understand now.

Re: Help needed with mathematical induction problem

Quote:

Originally Posted by

**HallsofIvy** 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"?

I should have said that I condensed my working a little in my first post. Yes, I did work out both sides and showed that they equal for my base step (using by parts).

I obviously don't understand factorials very well! But this has helped a lot. Thanks!

CP