# Binomial Series

## Homework Statement ## The Attempt at a Solution

Is there any difference between the above expression and ?

Is there any relation between these two?

VietDao29
Homework Helper

## Homework Statement Are you sure there are up to 2 sigma signs in that expression? By the way, you mean $$C_r^n$$ right?

If there's just one sigma, then $$\sum_{0 \le r < s \le n} (C_r^n + C_s^n)$$ is different from $$\sum_{r = 0}^n \sum_{s = 0}^n (C_r^n + C_s^n)$$.

In the first sum $$\sum_{0 \le r < s \le n} (C_r^n + C_s^n)$$, r, and s can take any value raging from 0 to n, but r must be less than s.

However, in the second sum: $$\sum_{r = 0}^n \sum_{s = 0}^n (C_r^n + C_s^n)$$, r, and s can take any value raging from 0 to n, no more requirement is needed.

So, in general, the second sum will have more terms than the first sum.

tiny-tim
Science Advisor
Homework Helper
Hi Abdul! The second one is roughly double the first, since it contains eg C1 + C2 but not C2 + C1.

hmm … what about all the terms such as C1 + C1 ? can you find an exact equation for the difference between the second and twice the first? Are you sure there are up to 2 sigma signs in that expression?

Yeah there are 2 sigma signs. 0<=r<s<=n is in between the two sigma signs.

can you find an exact equation for the difference between the second and twice the first?

Does that equate to ?

tiny-tim
Science Advisor
Homework Helper
Yes, except i'd call it 2 ∑Cr ok now write ∑∑ (Cr + Cs) over all r and s in terms of ∑Cr (try it first with an easy small number for n, like n = 3, if you're stuck)

Thanks!.... I got the answer 