What Are the Key Insights of the Binomial Series Homework Statement?

[zorro]

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 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]

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? ​

 

[zorro]

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]

Yes, except i'd call it 2 ∑C_r

ok now write ∑∑ (C_r + C_s) over all r and s in terms of ∑C_r

(try it first with an easy small number for n, like n = 3, if you're stuck)

 

[zorro]

Thanks!... I got the answer