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.

Question

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.

Ramesh V · Answer

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-1C_rx^r + .............

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

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

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

sum to n terms(u₀,u₁,u₂,...u_n-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)x^n-1 +⁽ⁿ⁺²⁾C_(n)xⁿ .....

sum to n terms(u₀,u₁,u₂,...u_n-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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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.

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.

1 Answers

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