Next page : The function the limit of (1+x/n)^n as n->infinity

Say you have borrowed on coin, of whatever domination, from a loan shark that takes 100% interest. After one year you will pay back 1+1=2 coins.

Now let us say the loan shark uses compounded interest to be compounded every month. That means that you have to pay 2.6 coins:

(Ok. This is a very stupid way to do the calculation, but it is how it is done. The question is why one don’t calculate the monthly interest rate as the 12^{th} root of 2…. but anyhow…)

Say we instead compound daily. This would give us

It looks like we are approaching some limit. So let us examine the limit

We first expand this using the binominal expansion

This gives us

Some cancelling and expansion later we have

Here the … in the numerators stand for some constants we don’t care about. This because all but the first term in each bracket will approach 0 as n goes toward infinity. We can do the same for the terms after the last …

This gives us

This will obviously be larger than 2 since we add positive numbers to the two first terms, 1+1.

Secondly, we have that 2!=2·1<2^{2}, and 2!=3·2·1<2^{2}, and so on, and we thus have that

And we know that

And thus that

I.e., the series converges to a value *e*, called Euler’s number. So, we have that

You can find a calculator calculating this with quite many more decimals here.

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