Prove that

(1 +

1 n

)n £ e < 3

.

Using Newton's formula (also called the Binomial Theorem)

æ ç è 1 + 1 n ö ÷ ø n   = 1 + n æ ç è 1 n ö ÷ ø + n(n-1) 2! æ ç è 1 n2 ö ÷ ø + n(n-1)(n-2) 3! æ ç è 1 n3 ö ÷ ø + ¼ £ 1 + 1 + 1 2! + 1 3! + ¼ £ e < 3.

(a) Which is larger: 1.000001^1000000 or 2?

It's possible to prove that every number in the form

(1+

1 a

)a

is greater than 2.

æ ç è 1+ 1 a ö ÷ ø a   = 1+a· 1 a +other positive terms > 2

for every integer a> 2. For a=1000000, the problem is solved.

(b) Which is larger: 100³⁰⁰ or or 300! (i.e.factorial 300)?

This is a bit more complex. I'll use the formula

(1+

1 n

)n = (

n+1 n

)n < 3     (1)

I will now use the last inequality to prove by induction that

n! > (

n 3

)n.

Clearly this is true for n=1 and 2 and so using the induction hypothesis that it is true for n:

(n+1)! = (n+1)n! £ (n+1) æ ç è n 3 ö ÷ ø n   = 3 æ ç è n+1 3 ö ÷ ø n+1   æ ç è n n + 1 ö ÷ ø n   .

So using (1) gives:

(n+1)! £ æ ç è n+1 3 ö ÷ ø n+1   .

The demonstration by induction is complete. In particular, for n=300, the formula solves the given problem.