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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.
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
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
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