Challenge Level

You are going to use a computer to work out the values of
$\sqrt{n}$, $n^n$ and $e^{-n}$ for a few values of the whole number
$n$.

Before you start, can you estimate values of $n$ beyond which each
function is larger than $100$, $1\,000\, 000$ or $1\,000\,
000\,000\, 000$? It would be great if you could specify as tight a
range as possible between which you know these values will be
exceeded.

Try them out on a computer. How close were you?

Excel can cope with numbers up to about $10^{308}$ (to a precision
of 16 places) What values of $n$ do you think will cause Excel to
break down for each function. Were you correct?

For large n, the factorial function can be approximated very well
with an expression of the following form:

$$n! \sim A n^{\pm n\pm 0.5} e^{\pm n}$$

Your challenge is to experiment using a calculator or
spreadsheet to find the constant $A$ and whether or not each $\pm$
is $+$ or $-$

Warning: You will need to think
about this task, as although the approximation gives a smaller and
smaller percentage error as $n$ increases, the absolute error
increases as $n$ increases!

When you find the correct form of the formula, can you numerically
calculate some large factorials to 8 significant figures? How large
a factorial will you be able to calculate?

You can now use this formula algebraically to perform calculations
beyond the limit of the computer. For example, you can estimate the
chance of getting $500$ heads if you toss a coin $1000$ times or
(equivalently) the number of ways of choosing $500$ people from a
group of $1000$ people.