How to find Euler's Constant

I am very new to MATLAB and need to program this formula:

gamma = lim m -> inf [sum k=1 to m((1/k) - ln(m + .5)]

I'm not sure how to simultaneously compute a limit and a summation simultaneously.

I tried: euler = limit(sum((1/k)-ln(m+.5)), k = 1..m), m, inf)

but it isn't working.

Answer 1

[John D'Errico]

To 50 digits, here is the value of the Euler-Mascheroni constant

0.57721566490153286060651209008240243104215933593992

Computing it is just a loop. Compute the sum of 1/k.

tic
format long g
C = 0;
for k = 1:1e9
  C = C + 1/k;
end
toc
C - log(k+0.5)

Elapsed time is 6.444622 seconds.
ans =
       0.577215664902138

So, after 6.4 seconds and 1 billion terms, I was pretty close. You won't be able to get 50 decimal digits that way though. It as been a while, but in the documentation for eulerGamma in my HPF tool, I had these comments:

% Uses the Bessel function method from:
% http://numbers.computation.free.fr/Constants/Gamma/gamma.pdf
%
% See also:
% http://en.wikipedia.org/wiki/Euler?Mascheroni_constant