Associated Topics || Dr. Math Home || Search Dr. Math

### Help Proving that the Sequence for the Natural Log Is Bounded

```Date: 09/15/2009 at 09:17:24
From: Rahul
Subject: It is about the derivation of the number 'e'

Let a(sub n) be the sequence [1 + (1/n)]^n.

I understand that the sequence is monotonically increasing.  But in
proving that it is also bounded, two of the steps remain unclear to
me.

This step is OK for me:

a(sub n) = 2 + (1/2!)[1 - (1/n)]
+ (1/3!)[1 - (1/n)][1 - (2/n)]
+ ... up to (n + 1) terms.

This step, however, I do not understand:

a(sub n) < 2 + (1/2!) + (1/3!) + ...... + (1/n!)

The reasoning for this is given to be that

1 - (1/n) < 1
1 - (2/n) < 1

That is how those terms containing n in the brackets behave.  (The
above inequalities also are understood because n being 1, 2, 3, ....
1/n  <= 1.)

Now, two times anything greater than 1 is greater than 2.  For example,
2*(1.5) = 3.

Two times anything less than 1 is less than 2.  For example,
2*(0.5) = 1; or 2*(0.9)= 1.8.

And 2*1 = 2.

So, the second step above is true for 2; but how do you generalize it?

```

```
Date: 09/16/2009 at 05:38:07
From: Doctor Ali
Subject: Re: It is about the derivation of the number 'e'

Hi Rahul!

Thanks for writing to Dr. Math.

Just note that when you accept that

1 - (1/n) < 1
1 - (2/n) < 1
...,

we can say that any product of the above expressions is still less
than one.

So, we can say that the original series are like

1 + 1/2! + 1/3! + 1/4! + ...,

where every term is multiplied by a number less than one.  Therefore,
it is less than the above series.

Does that make sense?

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search