# Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

Prove that $(1 + \frac{1}{n})^n \leq e < 3$.

Which is larger:

(a) $1.000001^{1000000}$ or $2$?

(b) $100^{300}$ or $300!$ (i.e. 300 factorial)?

Think about the Binomial Theorem.

Congratulations to Federico Poloni from Casirate d'Adda (Italy) for the following solution.

Prove that $(1 + \frac{1}{n})^n \leq e < 3$.

Using Newton's formula (also called the Binomial Theorem) $$\eqalign{ \left(1 + \frac {1}{n}\right)^n &=& 1 + n\left(\frac{1}{n}\right) + \frac{n(n-1)}{2!}\left(\frac{1}{n^2}\right) + \frac{n(n-1)(n-2)}{3!}\left(\frac{1}{n^3}\right)+ \cdots \\ &\leq & 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots \\ &\leq & e < 3. }$$ (a) Which is larger: $1.000001^{1000000}$ or $2$?

It's possible to prove that every number in the form $(1+\frac{1}{a})^a$ is greater than $2$. \[\left(1+\frac{1}{a}\right)^a = 1+a\cdot \frac{1}{a}+\mbox{other positive terms}> 2\] for every integer $a> 2$. For $a=1000000$, the problem is solved.

(b) Which is larger: $100^{300}$or $300!$ (i.e. $300$ factorial)?

This is a bit more complex. I'll use the formula $(1+ \frac{1}{n})^n = (\frac{n+1}{n})^n < 3 \quad (1)$

I will now use the last inequality to prove by induction that $ n!> (\frac{n}{3})^n.$ Clearly this is true for $n=1$ and $2$ and so using the induction hypothesis that it is true for $n$:

$(n+1)! = (n+1)n!
\leq(n+1)\left(\frac{n}{3}\right)^n$

$=
3\left(\frac{n+1}{3}\right)^{n+1} \left(\frac{n}{n +
1}\right)^n$.

So using (1) gives: $$ (n+1)! \leq \left(\frac{n+1}{3}\right)^{n+1}.$$ The demonstration by induction is complete. In particular, for $n=300$, the formula solves the given problem.

Why do this problem?

The numerical examples should prompt learners to formulate and prove a more general statement and then apply it to these special cases. Going from the particular to the general in problem solving is an important skill for a mathematician.

Inequalities play a big role in advanced mathematics and mathematical research and learners in school will benefit from experience of working with inequalities.

They need to know the Binomial Theorem and the formula for the exponential series and then the problem gives experience of applying these formulae and of proof by mathematical induction.

Possible approach

The first part could be a lesson starter or homework in preparation for a lesson or you could do the first part as a class and set the two numerical examples to be done independently.

Key questions

How do the numerical examples relate to $(1 +\frac{1}{n})^n$?