Gregory Series

Author

Eric W. Weisstein
September 10, 2008

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

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

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

Series

4 (∑_k^∞)((((-1))^(k+1))/(2 k-1))==π

4Sum[((((-1))^(k+1))/(2k-1)),{k,∞}]

(∑_k^∞)((((3^k)-1) ζ(k+1))/(4^k))==π

Sum[(((3^k)-1)/(4^k))Zeta[k+1],{k,∞}]

4 (∑_k^N)((((-1))^(k+1))/(2 k-1))==(((-1))^N) ((PolyGamma2[0, (N/2)+(1/4)])-(PolyGamma2[0, (N/2)+(3/4)]))+π

V6

4Sum[((((-1))^(k+1))/(2k-1)),{k,N}]

FullSimplify[%]

V7

4Sum[((((-1))^(k+1))/(2k-1)),{k,N}]

FunctionExpand[%,N∈Integers&&N>0]

(∑_(k=0)^∞)((EulerE[2 k])/((4^k) (N^(2 k+1))))~(PolyGamma2[0, (N/2)+(3/4)])-(PolyGamma2[0, (N/2)+(1/4)])

Sum[((EulerE[2k])/((4^k)(N^(2k+1)))),{k,0,∞}]

Sum[((EulerE[2k])/((4^k)(N^(2k+1)))),{k,0,∞}]//InputForm

N[{Sum[((EulerE[2k])/((4^k)(N^(2k+1)))),{k,0,130}],(-PolyGamma[0,(1/4)+(N/2)]+PolyGamma[0,(3/4)+(N/2)])}/.N->30,20]

N[{Sum[((EulerE[2k])/((4^k)(N^(2k+1)))),{k,0,130}],(-PolyGamma[0,(1/4)+(N/2)]+PolyGamma[0,(3/4)+(N/2)])}/.N->30,20]

N[With[{N=30},Table[((EulerE[2k])/((4^k)(N^(2k+1)))),{k,0,130}]],20]

N[With[{n=30,k=130},((EulerE[2k])/((4^k)(n^(2k+1))))],100]

Partial Series

FullSimplify[ps[n],n∈Integers&&n>0]

Series[(PolyGamma[0,(1/4)+(N/2)]-PolyGamma[0,(3/4)+(N/2)]),{N,∞,10}]

Table[-EulerE[2n]/4^n(((1/N))^(2n+1)),{n,0,5}]

With[{N=5 10^6},Table[-EulerE[2n]/4^n(((1/N))^(2n+1)),{n,0,5}]]

Series[Sec[x],{x,0,10}]

2Table[EulerE[2n],{n,0,5}]

ps[n_]:=4Sum[((((-1))^(k+1))/(2k-1)),{k,n}]

s="3."<>StringDrop[StringJoin@@ToString/@(Transpose[RealDigits[#,10,80][[1]]&/@{ps,Pi}]/.{{x_,x_}:>x,{x_,y_}:>"*"}),1]

Print/@{N[ps[5 10^6],80],N[Pi,80],s};

First/@StringPosition[s,"*"]