Binomial Series

Author

Eric W. Weisstein
January 28, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/BinomialSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BinomialSeries.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Special Cases

(∑_(k=0)^n)(x^k) (Binomial[n, k])==(((x+1))^n)

Sum[Binomial[n,k]x^k,{k,0,n}]

(∑_(k=0)^∞)(((((-x))^k) (Pochhammer[-r, k]))/(k!))==(((x+1))^r)

Sum[((Pochhammer[-r,k])/(k!))(-x)^k,{k,0,∞}]

Simplify/@Sum[((Pochhammer[-r,k])/(k!))(-x)^k,{k,0,3}]

(∑_(k=0)^∞)(((x^k) (Pochhammer[r, k]))/(k!))==(((1-x))^(-r))

Sum[((Pochhammer[r,k])/(k!))x^k,{k,0,∞}]

Sum[((Pochhammer[r,k])/(k!))x^k,{k,0,3}]

Sum[((Pochhammer[1/2,k])/(k!))x^k,{k,0,4}]

(∑_(n=0)^∞)(((2 n-1)!!)/((2 n)!!))==(1/(Sqrt[1-x]))

Sum[(((2n-1)!!)/((2n)!!))(x^n),{n,0,∞}]

(∑_(n=0)^∞)(((-1))^n) (Binomial[-(1/2), n]) (x^n)==(1/(Sqrt[1-x]))

Sum[(((-1))^n) Binomial[-(1/2),n](x^n),{n,0,∞}]

Series

Series[(1/(Sqrt[1-x])),{x,0,5}]

SeriesCoefficient[(1/(Sqrt[1-x])),{x,0,n}]

FullSimplify[%]

FunctionExpand[(((-1))^n) Binomial[-(1/2),n]]

FunctionExpand[Pochhammer[1/2,k]]

GeneratingFunction[(((-1))^n) Binomial[-(1/2),n],n,x]

GeneratingFunction[(Piecewise[{{(((-1))^n) Binomial[-(1/2),n], n≥0}, {0, True}}]),n,x]

GeneratingFunction[(((((-1))^n) (Sqrt[π]))/(Gamma[(1/2)-n] Gamma[1+n])),n,x]

GeneratingFunction[Pochhammer[1/2,n],n,x]

Continued Fraction

Table[((Ceiling[n/2](Ceiling[n/2]-(-1)^n k))/(n(n+1)))x,{n,3}]

TimeConstrained[1/(1-k x/(1+ContinuedFractionK[((Ceiling[n/2](Ceiling[n/2]-(-1)^n k))/(n(n+1)))x,1,{n,∞}])),120]//Timing

With[{k=.5,x=.3},{1/(1-k x/(1+ContinuedFractionK[((Ceiling[n/2](Ceiling[n/2]-(-1)^n k))/(n(n+1)))x,1,{n,20}])),(1+x)^k}]

With[{k=2.2,x=.7},{1/(1-k x/(1+ContinuedFractionK[((Ceiling[n/2](Ceiling[n/2]-(-1)^n k))/(n(n+1)))x,1,{n,20}])),(1+x)^k}]

(((1+x))^n)==(1/(1-((nx)/(1+((((1·(1+n))/(1·2))x)/(1+((((1·(1-n))/(2·3))x)/(1+((((2(2+n))/(3·4))x)/(1+((((2(2-n))/(4·5))x)/(1+((((3(3+n))/(5·6))x)/(1+…))))))))))))))