TITLE: Sum of exponential terms and binomial
QUESTION [5 upvotes]: I would like to calculate the following expression with large $m$:
$$\sum^{m}_{q=1} \frac{(-1)^{q+1}}{q+1} {{m}\choose{q}}  e^{-\frac{q}{q+1}\Gamma}.$$
But, due to the binomial, the computer gets stuck when $m$ grows large.
In this problem we have that $\Gamma > 0$.
I am trying to find a simplification or a way around, but I didn't find anything that could help me.
Can anyone give me some hints?

REPLY [2 votes]: This computation can be done numerically without a problem up to $m=2^{13}$.  It took about 15 seconds of cpu time using Mathematica on my 2 year old Mac laptop for $m=2^{13}$.
Here is an example of the Mathematica code:
s[m_, g_] := Sum[ (-1)^(q+1)/(q + 1) Binomial[ m , q] Exp[ - q g/(q + 1)], {q, 1, m}];
Print[ Timing[ s[10000, N[1/4, 10000]]//N]; 

The output for the program above is {27.7445,0.999574} indicating that it took 27 seconds to compute the answer.  Note that ${1000\choose 500}$ has about 3000 digits, so the program used 10000 digits of precision.  The running time is order $m^3$.    
The answer is usually close to 1 when $0<q<1$ and $m> 2^{10}$.  

I wrote the code in Python and got the same result for $m=2^{13}$ and $q=1/4$. 
from mpmath import mp

mp.dps =5000;

m = 2**13;

mp.pretty = True

rS = mp.mpf(0);

g = mp.mpf(1)/mp.mpf(4);

for q in range(1, m+1):
    rS = rS + mp.mpf((-1)**(q+1))* mp.binomial(m, q)/(q+1)*mp.exp(-q*g/(q+1));


mp.nprint(rS, 10)