$(1+\frac{1}{n})^{n} = 1$ for large numbers on calculator

Question

I've been experimenting with $(1+\frac{1}{n})^{n}$ on my calculator. Up until around $n=10^{10}$, I get approximately Euler's number. At some point after $n=10^{10}$, say, $n=10^{20}$, I just get a value of 1. Why is this? Why doesn't it continue tending towards Euler's number?

Answer 1

You are discovering the machine epsilon of your calculator. It does not store numbers with infinite precision. There is some smallest number $\epsilon$ for which $1+\epsilon \gt 1$. Anything smaller rounds off $1+\epsilon$ to exactly $1$. A large section of numerical analysis is devoted to avoiding this sort of problem. Your tests show it is somewhere between $10^{-10}$ and $10^{-20}$ and a little more experimentation can pin it down better. A helpful, though not precisely right, way to think of it is that the calculation is done exactly, then rounded off to some number of significant figures. Here that is between $10$ and $20$. There may be more significant figures than are shown in the display.