Let


$\theta(x)=\begin{cases}
0 & \text{ if } x<0 \\ 
1 & \text{ if } x\ge 0 
\end{cases}$


Do you know any way to calculate this number:


$$\sum\_{r=493701}^{506199}\sum\_{k=0}^{100}(-1)^k\frac{\binom{100}{k}\binom{r-10001k+99}{99}\theta(r-10001k+99)}{\binom{r+99}{r}}$$
or estimate it within 0.01. Gap and wolfram failed to calculate it. Also [octave](https://www.gnu.org/software/octave/) seems to be unable to calculate it.



Calculating a sum including large numbers
By comments after adding mpmath package to python one can run this program to see the result:



```
from mpmath import *
mp.dps=50

def th(x):
    if x<0:
        return 0
    else:
        return 1


def f(r,k):
    a=(-1)**k*binomial(100,k)*binomial(r-10001*k+99,99)*th(r-10001*k+99)
    a= a/binomial(r+99,r)
    return a

from timeit import default_timer as clock
time0 = clock()

s=0
for r in range(493701,506200):
    for k in range(0,101):
        s=s+ f(r,k)
time1 = clock()
print 'The result is'
print s
print 'computed in ', round(time1-time0,3), 'sec'


"""
Result:
0.0000000013060480365934635751811931267930306865307928

"""

```

mpmath is available in Ubuntu repository. Also one can write a C++ program using [GMP](https://gmplib.org/): <http://rextester.com/TGS53335> (this can be run only in a local computer with gmb installed and `-lgmb` library introduced to the compiler).


Gap can calculate to a fraction but cannot show the decimal representation.


