Proof of a Bounded Series using the Binomial Theorem

Date: 6 Jun 1995 23:53:53 -0400
From: barr
Subject: prove (1+1/n)^n is bounded

Please help me prove that (1+1/n)^n is bounded from above.

I was told to begin by proving:
1+r+r^2+...+r^n = (1-r^(n+1))/(1-r) 

for any r, [0,r],[r,1)

Could I do this simply by multiplying both sides by (1-r)?

Next I must use the binominal theorem to show that:

(1+1/n)^n is less than or equal to 1+{1+(1/2)+(1/2)^2+...+(1/2)^(n-1)}

I can't figure out how to expand the binomial in such a way to show 
this.  Even if I did, how would this prove that (1+1/n)^n is bounded 
from  above?

Thank you for your help,
Ken Barr

P.S.  This being our first week of Calculus, I am unfamiliar with 
differention, integration, etc.

Date: 8 Jun 1995 13:03:09 -0400
From: Dr. Ken
Subject: Re: prove (1+1/n)^n is bounded

Hello there!

          Could I do this simply by multiplying both sides by (1-r)?

Yes, that's what I would do.  The left side will then telescope to look just
like the right side.

          ... how would this prove that (1+1/n)^n is bounded from above.

Yes, this is kind of tricky.  The binomial theorem says that the left side is

(1 + 1/n)^n = 1 + [n:1](1/n) + [n:2](1/n)^2 + [n:3](1/n)^3 +...+[n:n](1/n)^n.

where the [n:1] and so on means the "choose" symbol, [n:k] = n!/(k!(n-k)!).
If n and k are integers, then [n:k] is also an integer.  See if you can
manipulate the right side of the equation to get the form you're looking
for.  Hint: how big are the coefficients [n:k]?

Once you've done that, look at what you've got.  (1+1/n)^n is less than or
equal to 1+{1+(1/2)+(1/2)^2+...+(1/2)^(n-1)}, and we can replace the
bracketed expression with what we got above and get
1+{(1-.5^n)/(1-.5)} = 1+{1-.5^n/.5} 
                    = 1+2(1-.5^n).

Now can you show that this thing is bounded?

-K

Date: 8 Jun 1995 22:15:55 -0400
From: barr
Subject: Prove (1+1/n)^n is bounded

Thanks for your quick response!  (This may be handy in the future!)  Our 
next-year's-calc-teacher explained "n to the n FEPL" to us which helped 
AND we realized that we were not setting out to show that the limit was e 
but simply that it HAD a bound.

Anyway, thanks again.  Have a nice summer,

Ken