TITLE: Why WolframAlpha Result is Different?
QUESTION [6 upvotes]: http://www.wolframalpha.com/input/?i=x%3D10%5E-15%3B+%28e%5Ex-1%29%2Fx
When i click Approximate Form in result, the result is:
1.11022

but when i click More Digit:
1.000000000000000500000000000000166666666666666708333333333333341666666666666668055555555555555753968

Why the number is different? (instead of showing more digit) and what is the actual solution?

REPLY [3 votes]: You are seeing exactly the exponential expansion in both cases, limited to the available precision, which is computed with a 64 bit double in the default case and a hundred digits in the "More Digits" case.
Let's try it by hand:  $$e^x \approx 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \dots$$
With a double: $1 + 10^{-15} + 10^{-30}/2 + \dots$ and we have 53 bits $\approx$ 17.46 decimal digits of accuracy, so we can drop the $x^2$ and higher terms.
Now, what is $1+10^{-15}$ in 64 bit floating point representation?  Well, it certainly can't represent it exactly.  Rather, we have a 53 bit binary number which tries to get as close as possible.  Note that $2^{-50}$ + $2^{-52}$ = $1.110223\dots \cdot 10^{-15}$, so that's the best we can do.
Subtract off $1$ and you're left with this as an error term.
The hundred-digit one is correct.  You can see the $10^{-15}$ term, the half-$10^{-30}$ term (5 and then a bunch of zeros), the cubic term--it's $1 over 6$ so you get 16666 in the decimal expansion, and then the quartic term comes in on top of it $6666666... + 0416666... \rightarrow 70833333$ etc..