Proof that a sum is bounded by given functions?

Hey there!

I'm really stuck on a question and i can't figure out how I should solve it.

I have a sum given by

Given the special case where

I have to prove that is logarithmically bounded by

And here's a *HINT* : Look at the terms

in

Well, I really don't get it...if someone could help me I would greatly appreciate!

ps. I'm glad I found thoses forums, I'll certainly become an active member, being a studient in computer sciences :P.

THANKS!

Re: Proof that a sum is bounded by given functions?

Prove the required claim by induction on m. This requires bounding from below and from above, as per the hint.

Re: Proof that a sum is bounded by given functions?

Thank you for your reply!

I kinda guessed it was by induction but didn't think about the bounding from below. (I know... i had a rough day)

But still I'm confused about the induction hypothesis :S

Normally, inductions are all about a base case ( ) and an inductive step (do we need to show for ?)

I'm really confused! Thank you!

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by

**Flexdec**
Given the special case where

I have to prove that

is logarithmically bounded by

Quote:

Originally Posted by

**Flexdec** I kinda guessed it was by induction but didn't think about the bounding from below. (I know... i had a rough day) But still I'm confused about the induction hypothesis :S

Normally, inductions are all about a base case (

) and an inductive step (do we need to show for

?)

Actually the is very well know similar inequality:

.

That gives you very quickly the RHS.

But I admit that I have not seen the LHS bound. So perhaps induction is the best approach.

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by

**Flexdec** But still I'm confused about the induction hypothesis :S

Normally, inductions are all about a base case (

) and an inductive step (do we need to show for

?)

Yes.

To be more explicit, prove P(m) for all by induction on m where P(m) is .

In the induction step, when you are proving P(m+1) from P(m), you know the lower and upper bounds on . Also, it is easy to get lower and upper bounds on by replacing the tail of the sum with its smallest or biggest term, respectively.

Re: Proof that a sum is bounded by given functions?

Ok so i give it a try. Given the special case , the sum is given by

Lets prove, by induction, the following logarithmic bounds

*edited ^

First, lets start with the base case

???? ????

well from there I get stuck, I think I did something wrong ...

Thank you for your help so far!!

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by

**Flexdec** Given the special case

, the sum is given by

Lets prove, by induction, the following logarithmic bounds

Why did you replace with ? The sum starts with 1.

Re: Proof that a sum is bounded by given functions?

I guess the hint messed with my head :P

So for base case it would be

right?

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by

**Flexdec** So for base case it would be

right?

Yes.

Re: Proof that a sum is bounded by given functions?

and from there...

I'm lost :S

Re: Proof that a sum is bounded by given functions?

Look again at post #5. You assume the induction hypothesis P(m), i.e., . From this assumption, you need to prove P(m+1), i.e., . Now, . The induction hypothesis gives you an upper and a lower bound on the first term of this sum, i.e., . As for the tail of , replace all terms with the smallest term to get a lower bound and with the largest term to get an upper bound. By adding the obtained lower bounds on and the tail you get a lower bound on , and by adding the obtained upper bounds on and the tail, you get an upper bound on . The two resulting inequalities are precisely P(m+1).

Re: Proof that a sum is bounded by given functions?

What do you mean by tail, all the terms except the first?

thanks!

Re: Proof that a sum is bounded by given functions?

Quote:

Originally Posted by

**Flexdec** What do you mean by tail, all the terms except the first?

Yes, by the "tail of " I refer to .

Re: Proof that a sum is bounded by given functions?

And one last thing!!!

when you say replace all terms you mean replacing all the terms of the tail right?

I can't figure out how it can equal the lower and upper bounds when added to bounds.

For example, for lower bound:

where T is the modified tail you're talking about. would T be like

this is what I understood..

Re: Proof that a sum is bounded by given functions?