Binomial Series

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  • #1
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Homework Statement



ue&space;of}\sum_{0\leq&space;r<&space;s\leq&space;n}\sum&space;(C_{r}&space;&plus;&space;C_{s}).gif



The Attempt at a Solution



Is there any difference between the above expression and
gif.latex?\sum_{r=0}^{n}\sum_{s=0}^{n}(C_{r}&plus;C_{s}).gif
?

Is there any relation between these two?
 

Answers and Replies

  • #2
VietDao29
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Homework Statement



ue&space;of}\sum_{0\leq&space;r<&space;s\leq&space;n}\sum&space;(C_{r}&space;&plus;&space;C_{s}).gif

Are you sure there are up to 2 sigma signs in that expression? By the way, you mean [tex]C_r^n[/tex] right?

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

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

However, in the second sum: [tex]\sum_{r = 0}^n \sum_{s = 0}^n (C_r^n + C_s^n)[/tex], 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.
 
  • #3
tiny-tim
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Hi Abdul! :smile:

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 ? :rolleyes:

can you find an exact equation for the difference between the second and twice the first? :smile:
 
  • #4
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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
gif.latex?\sum_{r=0}^{n}(C_{r}&space;&plus;&space;C_{r}).gif
?
 
  • #5
tiny-tim
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Yes, except i'd call it 2 ∑Cr :smile:

ok now write ∑∑ (Cr + Cs) over all r and s in terms of ∑Cr :wink:

(try it first with an easy small number for n, like n = 3, if you're stuck)
 
  • #6
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Thanks!.... I got the answer :smile:
 

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