So what happens here is that we write down four equal factors--one plus X over four times one plus X over four,
times one plus X over four, times one plus X over four. And then we look what this is going to turn into.
For the first box we need everything that has no X whatsoever. This is just one times one times one times one.
Anything else would have an X in it. So this becomes one. For the next box, we need everything that has precisely one X in it.
For instance, X over four times one times one times one. One times X over four times one times one.
One times one times X over four times one, and there's a fourth one: One times one times one times X over four.
We get four of these. Four times X over four, which is one times X. Four X squared.
We have X from the first two parentheses, but one from parentheses number three and parentheses number four.
X from the first, one from the second, X from the third, one from the fourth.
Or X from the first, one from the second, one from the third, and X from the fourth.
Or one from the first, X from the second, X from the third, one from the fourth, one from the first, X from the second,
one from the third, and X from the fourth. And the final one, one from the first, one from the second, X from the third, X from the fourth.
Six options to form something that contains X squared. Six times X over four times X over four.
We can cancel a factor of two, which leaves us with three over eight.
To get X to the third power, we need to pick X from three parentheses, and one from the other.
We can pick X from the first three parentheses, and the one from the last, or X from parentheses number one, number two,
the one from the third, X from the last, X from the first parentheses, one from the second, X from the third,
X from the fourth, and one from the first, X from the second, X from the third, X from the fourth.
So, four ways of doing this, four times X over four times X over four, times X over four, which leaves us with X to the third power
over 16. One over 16. And the final one: We need four factors of X.
You have to pick this and this and this and this. That's it. Only one option for doing that.
Which leaves us with X over four times X over four times X over four times X over four.
X to the fourth power divided by 256. One over 256.
So this is a crude approximation to the exponential function, and we find a way of expressing this with powers of X.
The beginning of a power series. Let me restate the general equation for the exponential function again.
E to the power of X is the limit as N goes to infinity of one plus X over N to the power of N.
We have now looked at the special case of N equals four. Hopefully, you can imagine that this also works out
for 40 or 400, 4,000, four billion, four whichever number. And the result is it's one plus X--
We have that already. One plus one X, plus X squared over two. You see we're close.
Four over eight would be one half. We've got three over eight, with that four here.
Plus X to the third power over six, plus X to the fourth power over 24, plus, and so on, up to infinity.
To prove this, you need a better method than we have used here; you need binomial coefficients.
But hopefully, you get the idea where this power series comes from.
And actually, it turns out that the six is the factorial of three; this 24 is the factorial of four, and so on.
So the general term would be X to the power of N divided by N factorial.
This also gives us a way of computing E itself. E equals E to the power of one.
So we simply set X to one, which means E equals one plus one plus one over two, plus one over six, plus one over 24,
plus, and so on, up to infinity. And now you can see how efficient this power series is.
One plus one, so we have two already, plus one half, is 2.5, plus one over six equals 2.6.
You see we're getting pretty quickly pretty close to 2.71 something. So if you actually want to computer Euler's number,
or the exponential function, you're using this power series.