Here’s an approximation to e by Richard Sabey that uses the digits 1 through 9 and is accurate to over a septillion digits. (A septillion is 1024.)
MathWorld says that this approximation is accurate to 18457734525360901453873570 decimal digits. How could you get an idea whether this claim is correct? We could show that the approximation is near e by showing that its logarithm is near 1. That is, we want to show
Define k to be 3^(2^85) and notice that k also equals 9^(4^42). From the power series for log(1 + x) and the fact that the series alternates, we have
where η is some number between 0 and 1/k. This tells that the error is extremely small because 1/k is extremely small. It also tells us that the approximation underestimates e because its logarithm is slightly less than 1.
Just how small is 1/k? Its log base 10 is around -1.8 × 10^25, so it’s plausible that the approximation is accurate to 10^25 decimal digits. You could tighten this argument up a little and get the exact number of correct digits.