Dirichlet Eta Function η(x) Calculator

Evaluate the alternating zeta series η(x)=Σ(-1)^(n-1)/n^x. Also shows the relation to the zeta function η(x)=(1-2^(1-x))ζ(x) and η(1)=ln2.

Input

Enter a real number x to evaluate the Dirichlet eta function η(x)=Σ(-1)^(n-1)/n^x, the alternating zeta series. Values of x at or below 0, including negatives, are allowed.

Exponent x

Any real number. Examples: 2, 0.5, -1

Result

η(2)

0.8224670334

η(x)=Σ(-1)^(n-1)/n^x, computed with a convergence-acceleration method.

ζ(2) (zeta function)

1.6449340668

factor 1 − 2^(1−x)

0.5

Plot of η(x)

Range x = -4 to 8. The orange dot marks your input.

Partial sums (naive alternating series)

Each term (-1)^(n-1)/n^x and its running total. The naive series converges slowly, so these are for reference.

n	n-th term	Partial sum
1	1	1
2	-0.25	0.75
3	0.11111111	0.86111111
4	-0.0625	0.79861111
5	0.04	0.83861111
6	-0.02777778	0.81083333
7	0.02040816	0.8312415
8	-0.015625	0.8156165
9	0.01234568	0.82796218
10	-0.01	0.81796218
11	0.00826446	0.82622664
12	-0.00694444	0.81928219

How it works

The Dirichlet eta function is the alternating zeta series η(x)=Σ(n=1 to ∞) (-1)^(n-1)/n^x.
It is tied to the Riemann zeta function by η(x)=(1-2^(1-x))ζ(x).
At x=1 the zeta function has a pole, but η stays finite with η(1)=ln2 (about 0.6931).
This tool uses a convergence-accelerated alternating-series algorithm, giving stable values across a wide real range including x at or below 0.
The naive partial sums Σ(-1)^(n-1)/n^x shown in the table converge slowly and are for reference; the reported η(x) uses the accelerated value.
Reference values: η(2)=π^2/12 (about 0.8225), η(4)=7π^4/720 (about 0.9470), η(0)=1/2.

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