= Euler evaluation of even zeta values
{c}
{title2=$\zeta(2r)=(-1)^{r+1}B_{2r}(2\pi)^{2r}/(2(2r)!)$}
The <Euler evaluation of even zeta values> turns a family of <series> into exact expressions using <Bernoulli numbers>, <factorials> and powers of $\pi$. For every <positive integer> $r$,
$$
\boxed{\sum_{n=1}^{\infty}\frac1{n^{2r}}=\zeta(2r)=\frac{(-1)^{r+1}B_{2r}(2\pi)^{2r}}{2(2r)!}}.
$$
Here $\zeta$ is the <Riemann zeta function>, defined by its <convergent series> at these arguments, and the <Bernoulli number> convention is $B_1=-1/2$. The alternating sign combines with the sign of $B_{2r}$ to give a positive <sum>.<NIST Digital Library of Mathematical Functions/25.6 Integer Arguments>[§25.6(i), equations 25.6.1–25.6.2: positive even arguments]{ref}
Substituting $B_2=1/6$, $B_4=-1/30$ and $B_6=1/42$ gives
$$
\boxed{\zeta(2)=\frac{\pi^2}{6},\qquad\zeta(4)=\frac{\pi^4}{90},\qquad\zeta(6)=\frac{\pi^6}{945}}.
$$
The first identity answers the <Basel problem>. It is surprising because the original <series> only mentions reciprocal squares of <integers>, while its exact <sum> involves the circle constant $\pi$. The <cotangent partial-fraction expansion> supplies a connection between the two: <trigonometric functions> encode equally spaced integer <poles>, whose expansion brings the reciprocal powers into view.
For an original numerical comparison, the first ten terms give $\sum_{n=1}^{10}n^{-2}\approx1.549767731$, while $\pi^2/6\approx1.644934067$. The omitted positive terms still contribute approximately $0.095166336$. For fourth powers, the ten-term <partial sum> is approximately $1.082036583$, only $0.000286650$ below $\pi^4/90\approx1.082323234$. Both are <convergent series>, but the second approaches its <limit> much faster.
\Image[/zeta-even-power-partial-sums.png]
{title=Reciprocal-square and reciprocal-fourth-power partial sums approach exact zeta values}
{description=Original computed <partial sums> for the first thirty terms. Each panel has its own vertical scale. Dashed horizontal lines mark $\pi^2/6$ and $\pi^4/90$; neither finite <partial sum> equals its limiting value.}
Exact evaluation is useful beyond obtaining more digits. It supplies an independent check on a numerical <summation>, exposes how <Bernoulli numbers> organize infinitely many identities, and makes the remaining <series truncation error> measurable without knowing any omitted terms individually. The <even integer> condition is essential to this formula: putting $r=3/2$ into it is not a way to evaluate $\zeta(3)$, since the required <Bernoulli number> index and <factorial> expression here are indexed by <positive integers>.
Historically, <Leonhard Euler> announced reciprocal-square and reciprocal-fourth-power evaluations in §2 of <On the sums of series of reciprocals by Leonhard Euler>. That paper was presented on 5 December 1735 and published in 1740, as the English translation's first-page note records. His §§3–8 develop a route through the <sine> <power series> and relationships between roots and <coefficients>. The <cotangent> calculation below is a concise modern presentation of the connection.<On the sums of series of reciprocals by Leonhard Euler>[English translation, p. 1, §2 and publication note; pp. 3–4, §§3–8: sine series, roots and coefficients]{ref}
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