Concrete mathematics : a foundation for computer science

Heisenberg may
have been here.
9.6 FINAL SUMMATIONS 467
This equation, incidentally, gives us a good approximation to y even when
n = 2:
y = Hz-ln2--i+&-&,+e = 0.577165...+~,
where E is between zero and &. If we take n = lo4 and m = 250, we get
the value of y correct to 1271 decimal places, beginning thus [171]:
y = 0.57721566490153286060651209008240243... . (9.90)
But Euler’s constant appears also in other formulas that allow it to be evaluated
even more efficiently [282].
Summation 3: Stirling’s approximation.
If f(x) = In x, we have f'(x) = 1 /x, so we can evaluate the sum of
logarithms using almost the same calculations as we did when summing reciprocals.
Euler’s summation formula yields
t
Ink = nl.nn-n+o-F
1 $k