Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Explanation?

Hi;

It is a nice result anyway is the correct attitude.

Do you use Maple alot? Do you like it?

I am going to take a break, see you later.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Maburo

Member

From: Alberta, Canada

Registered: 2013-01-08

Posts: 287

Re: Explanation?

Hello. This is Ma123 on my new account with my preferred username. I haven't been on the forum for a long time. In my free time I worked out a proof by induction for the formula

. My first actual proof on something I have never previously seen

To do it, I had to create a function of two variables and prove that the nth difference of perfect powers of n is n!. Then I used that result to show that the nth difference can be expressed by the alternating sum, thus proving that n! is equal to the alternating sum. If anybody would like to see, I could practice my LaTeX skills and write it up on here some time

Also, I have been reading the forum for quite some time now. I really enjoy it, so I will probably be spending some time here. I guess this is sort of an introduction, so hello all!

---

"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Explanation?

Hi;

No problem, I knew who you were. Welcome to the forum.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: Explanation?

Welcome to the forum!

Why not make another Introduction thread?

Maburo

Member

From: Alberta, Canada

Registered: 2013-01-08

Posts: 287

Re: Explanation?

Here is the proof:

Consider the difference pyramid discussed earlier in the thread. Each row is the difference between consecutive terms of the previous row. We will define a function to describe the differences:

. Here

represents the number of times we have taken the difference,

is a number in a row, and

is the exponent. By this, we can define the first row (the 0th difference) with

. Thus, by our definitions,

. This should make sense because

is the first difference between two consecutive powers of n.

We can expand this:

Now we can apply the same process for further differences:

This is where we use induction: we want prove that the nth difference is equal to n!

To do this we need to assume two things:
First, we assume that the nth difference is, in fact, equal to n!, and we need to assume that after we reach a row of n!, the next difference is 0.

Assume:

Now we prove this for

:

This was just the first part of my proof, showing that the nth difference of perfect powers of n is equal to n!. I will use this to prove that n! can be expressed as an alternating sum. I have to go for now, so I will post that part later. Bye for now! And let me know if I made any errors or did anything improperly. I didn't catch any mistakes, but I'm sure you guys can if I made any.

---

"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein

Maburo

Member

From: Alberta, Canada

Registered: 2013-01-08

Posts: 287

Re: Explanation?

Now that we have shown that the nth difference in our pyramid gives a row of

, and that every row after that is all

, we will show that

can be expressed as an alternating sum:

Remember the definition:

   and   

We can continue this and notice a pattern:

and

So in general:

What we want to prove is that:

To do so, we will prove that

  and then look at the case where

and we already proved that

  when 

Again we use induction:

We must assume that

Then

Expanding:

We see pairs of

and

Now we rewrite our previous expansion as:

Every k has been replaced with k+1, thus proving that

Now for the best part! We shall look at the case where k=n:

becomes

which we proved was equal to

Substituting n for k in our sum:

Now, x is arbitrary; it does not matter where in a sequence of perfect powers of n we begin taking differences, so we will let x=0:

That is my proof Let me know if you guys see any errors (maybe I am completely wrong). Also, I apologize if it is a messy proof either due to my lack of LaTeX skills or lack of organization. Off to bed now! Bye.

---

"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein

Maburo

Member

From: Alberta, Canada

Registered: 2013-01-08

Posts: 287

Re: Explanation?

Hello again. Here it is, written with WriteLaTeX. This version has been cleaned up. It is much more organized and includes a lot more explanation than my posts on the forum.

Proof

---

"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein