Math Is Fun Forum

theshire

Member

Registered: 2022-08-27

Posts: 3

Two sequences and the exponential function

It all started out some years ago when I read how the number e was introduced / defined. I thought to myself: Hey, why not explore the sequences

for some real number r.

Let

such that

. Define

Then for some

Thus

Therefore

Looking at the sequence of partial sums

:

Hence

So

Therefore

Now pick some positive real number r. Define

Then for all integers n such that

So if you pick some

, then there exists an integer N such that for all integers greater than N:

Hence

Define

for

.

So finally we get

Last edited by theshire (2023-01-04 13:54:50)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,618

Re: Two sequences and the exponential function

hi theshire

I'm seeing a lot of LaTex errors in this.  I hope you don't mind but I've had a go at editing some but I'm worried I might have changed your proof in the process.  Please have a look and see if it's ok. Also ask if you are unsure of the Latex that works for MIF. Not all feaures of the coding do, I'm sorry. What did you want list to do?  Some frac errors towards the end.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob