Sum of combinations from k to n

[Ediliter]

I have been trying to figure out a formula for the sum of combinations. For example:

$\sum$ⁿ_k=0($\frac{n}{k}$) = 2ⁿ

But what if you want to sum from any arbitrary k, like 4? I've tried looking at Pascal's triangle for nice values of n and k, but haven't been able to see a pattern. I would really appreciate any help with this. I want to apply this to combinations for large n, which are impractical to compute.

Thank you in advance.

 

[Stephen Tashi]

I don't know any nice formula for $\sum_{k=0}^m \binom{n}{k}$ Your question made me curious and I searched the web. It apparently doesn't know a nice formula either. Perhaps if you give an example of the kind of computation you are trying to do, someone will see a way to compute the result - at least compute it on a computer.

 

[bpet]

The sum could be expressed in terms of the incomplete beta function, e.g. using the cdf of the binomial distribution with p=1/2.

 

[mXSCNT]

For large n the binomial distribution is approximated by a normal distribution, so if you only want a close approximation you could use that.