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# Let u_n = coefficient of x^n in (1-x)^{-p}, p is an integer Find the sum to N-terms of the series {u_0,u_1...} for a) p=1 b) p=2 c) p=3 Genralize the procedure.

12 years ago

These binomial expansions are infinite

The general series is :

(1-x)-p =1+ px+ p(p+1)/2! x2 + p(p+1)(p+2)/3! x3 + ............ + p+r-1Cr xr + .............

for p=1   (1-x)-1 = 1+x+x2+x3+x4+ ......  +xn-1 + .....

sum to n terms(u0,u1,u2,...un-1) is 1+1+1+...(n) terms  ,  so sum is n

for p=2   (1-x)-2 = 1+2x+3x2+4x3+5x4+ ......  +(n)xn-1 + .....

sum to n terms(u0,u1,u2,...un-1) is 1+2+3+...+n  , so sum is n(n+1)/2

for p=3   (1-x)-2 = 1+3x+6x2+10x3+ ......  +(n+1)C(n-1)xn-1 +(n+2)C(n)xn .....

sum to n terms(u0,u1,u2,...un-1) is 2C0+3C1+4C2+5C3+..........+ (n+1)C(n-1)     so sum is

i.e.,  = 1 +  Σ (n+1)C(n-1)

=  1 + [( Σn2 + Σn) /2]

sum to n terms for p=3       = ( 2n3 + 3n2+ n +6)/6