Alternating Series

Author

Eric W. Weisstein
January 3, 2006

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

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

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

Examples

(∑_(k=0)^∞)((((-1))^k)/(k!))==(1/e)

V5

Sum[((((-1))^k)/(k!)),{k,0,∞}]

V7

Sum[((((-1))^k)/(k!)),{k,0,∞}]

(∑_k^∞)((((-1))^(k-1))/k)==log(2)

V5

Sum[((((-1))^(k-1))/k),{k,∞}]

V7

Sum[((((-1))^(k-1))/k),{k,∞}]

(∑_k^∞)((((-1))^(k-1))/(k^2))==((π^2)/12)

V5

Sum[((((-1))^(k-1))/(k^2)),{k,∞}]

V7

Sum[((((-1))^(k-1))/(k^2)),{k,∞}]

(∑_k^∞)((((-1))^(k-1))/(k^3))==((3 (Zeta[3]))/4)

V5

Sum[((((-1))^(k-1))/(k^3)),{k,∞}]

V7

Sum[((((-1))^(k-1))/(k^3)),{k,∞}]

No Known Closed Form

(∑_(n=1)^∞)(((-1))^(n+1)) (e-(((1+(1/n)))^n))

V5

Sum[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,∞}]

V7

Sum[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,∞}]

SumConvergence[(((-1))^(n+1))(E-(((1+(1/n)))^n)),n]

Sequence

Table[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,10}]

Table[((((n+1))^n)/(n^n)),{n,10}]

Numeric

NSum[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,∞}]

NSum[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,∞},WorkingPrecision->100,PrecisionGoal->100]

NSum[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,∞},WorkingPrecision->500,PrecisionGoal->100]

NSum[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,∞},WorkingPrecision->100,Method->Integrate]

SequenceLimit[FoldList[Plus,0,N[Table[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,100}],1000]]]

SequenceLimit[FoldList[Plus,0,N[Table[(((-1))^(n+1))(E-(((1+(1/n)))^n)),{n,500}],1000]]]

Transformed

FullSimplify[Plus@@({#,#/.n->n+1}&/@((((-1))^(n+1))(E-(((1+(1/n)))^n))))/.n->2n-1,n∈Integers]

V5

Sum[(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n)),{n,∞}]

V7

Sum[(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n)),{n,∞}]

NSum[(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n)),{n,∞}]

Plot[(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n)),{n,1,100},PlotStyle->Red]

(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n))/.n->10^20`100

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

FoldList[Plus,0,%]

Limit[#,n->∞]&/@{(((1+(1/(2 n))))^(2 n)),(((1+(1/(-1+2 n))))^(-1+2 n))}

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

(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n))

Limit[(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n)),n->∞]

Sum[Binomial[2n,k]((((1/(2n))))^k),{k,0,2n}]//Simplify

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

expr=FullSimplify[Sum[Binomial[2n,k]((((1/(2n))))^k)-Binomial[2n-1,k]((((1/(2n-1))))^k),{k,0,2n}],n>0&&n∈Integers]

FullSimplify[Binomial[2n,k]((((1/(2n))))^k)-Binomial[2n-1,k]((((1/(2n-1))))^k),0≤k≤2n]

Simplify[Subtract@@{expr,(((1+(1/(2 n))))^(2 n))-(((1+(1/(-1+2 n))))^(-1+2 n))},n>0&&n∈Integers]

N[Sum[expr,{n,300}],10]

Sum[(4^(-n)) (((2+(1/n)))^(2 n))+(2^(-1+2 n)) (((1-2 n))^(-2 n)) (n^(-1+2 n))-(4^n) ((((n/(-1+2 n))))^(2 n)),{n,∞}]

Sum[(4^(-n)) (((2+(1/n)))^(2 n))+(2^(-1+2 n)) (((1-2 n))^(-2 n)) (n^(-1+2 n))-(4^n) ((((n/(-1+2 n))))^(2 n)),{n,∞}]

Transformed by Paul Abbott

Using ((((1+(1/n)))^n)==(e^(n log(1+(1/n)))))

V5

(1/2)Log[2]E+NSum[((StirlingS1[p+l,l])/((p+l)!))(2^((1-p)-1))Zeta[p],{l,0,∞},{p,2,∞}]

(1/2)Log[2]E+Sum[((StirlingS1[p+l,l])/((p+l)!))(2^((1-p)-1))Zeta[p],{l,0,∞},{p,2,∞}]

(1/2)Log[2]E+Sum[((StirlingS1[p+l,l])/((p+l)!))(2^((1-p)-1))Zeta[p],{p,2,∞},{l,0,∞}]

V7

(1/2)Log[2]E+Sum[((StirlingS1[p+l,l])/((p+l)!))(2^((1-p)-1))Zeta[p],{l,0,∞},{p,2,∞}]