Series Multisection

Author

Eric W. Weisstein
December 26, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Binomial

V7

Sum[Binomial[n,t+s m],{m,0,n}]

TimeConstrained[(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}],300]//Timing

s==1, t==0

With[{s=1,t=0},Sum[Binomial[n,t+s m],{m,0,∞}]]

Table[%,{n,10}]//Simplify

With[{s=1,t=0},(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}]]

s==2, t==0

With[{s=2,t=0},Sum[Binomial[n,t+s m],{m,0,∞}]]

Table[%,{n,10}]//Simplify

Simplify[Sum[Binomial[n,2 m],{m,0,n}],n∈Integers]

Table[%,{n,10}]//Simplify

Table[Sum[Binomial[n,2 m],{m,0,n}],{n,10}]

With[{s=2,t=0},(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}]]

Table[%,{n,10}]//Simplify

s==2, t==1

With[{s=2,t=1},Sum[Binomial[n,t+s m],{m,0,∞}]]

Table[%,{n,10}]//Simplify

With[{s=2,t=1},(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}]]

Table[%,{n,10}]//Simplify

s==3, t==0

With[{t=0,s=3},Sum[Binomial[n,t+s m],{m,0,∞}]]

Table[%,{n,10}]//Simplify

FullSimplify[With[{t=0,s=3},(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}]
],n∈Integers&&n>0]

Table[%,{n,10}]//Simplify

s==3, t==1

With[{t=1,s=3},FullSimplify[Sum[Binomial[n,t+s m],{m,0,n}],n∈Integers&&n>0]]

Table[%,{n,10}]//Simplify

FullSimplify[With[{t=1,s=3},(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}]
],n∈Integers&&n>0]

Table[%,{n,10}]//Simplify

s==3, t==2

With[{t=2,s=3},FullSimplify[Sum[Binomial[n,t+s m],{m,0,n}],n∈Integers&&n>0]]

Table[%,{n,10}]//Simplify

FullSimplify[With[{t=2,s=3},(1/s)Sum[Cos[((π(n-2t)j)/s)](2^n)((Cos[((π j)/s)])^n),{j,0,s-1}]
],n∈Integers&&n>0]

Table[%,{n,10}]//Simplify