= Basel problem
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{title2=$\sum_{n=1}^{\infty}n^{-2}=\pi^2/6$}
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The <Basel problem> asks for the exact <sum> of the reciprocal squares of the <positive integers>. The answer is
$$
\boxed{\sum_{n=1}^{\infty}\frac1{n^2}=\zeta(2)=\frac{\pi^2}{6}}.
$$
The <integral test> proves that the <series> converges, but does not identify this exact value. The <Euler evaluation of even zeta values> does: the equally spaced <poles> of <cotangent> connect reciprocal <integer> powers to <Bernoulli numbers> and $\pi$. This separates three different questions: whether a <sum> exists, how well a <partial sum> approximates it, and what its exact value is.
For example, the ten-term <partial sum> is about $1.549768$, while the exact value is about $1.644934$. The <zeta series truncation error> bounds the missing amount between $1/11$ and $1/10$; it is much larger than the first omitted term $1/121$. More terms improve the approximation, but a numerical coincidence with $\pi^2/6$ would not prove the identity.
\Image[/zeta-even-power-partial-sums.png]
{title=The Basel series converges slowly compared with the reciprocal-fourth-power series}
{description=The left panel shows original computed reciprocal-square <partial sums> approaching $\pi^2/6$. The right panel compares the faster convergence of reciprocal fourth powers. Each panel has its own vertical scale.}
In <On the sums of series of reciprocals by Leonhard Euler>, §2 identifies the reciprocal-square <sum> with one-sixth of the squared circumference of a unit-diameter <circle>. The translation's publication note dates the presentation to 5 December 1735 and publication to 1740.<On the sums of series of reciprocals by Leonhard Euler>[English translation, p. 1, §2 and publication note]{ref} The <cotangent coefficient proof of the even zeta values> gives a short modern outline, while <zeta series truncation error> addresses the distinct approximation question.
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