Basel problem

Parametric equations

The Basel problem asks for $\sum _{n=1}^{\infty }\frac{1}{n^2}$. Euler proved that the sum is $\frac{{\pi }^2}{6}$. An infinite product for $\sin x$, or a split into areas, is one way to tie that series to the square of $\pi$.

The partial sum $S_N=\sum _{n=1}^N\frac{1}{n^2}$ increases and approaches $\frac{{\pi }^2}{6}$.

Interaction

• Number of terms N: add terms from a few toward many, and watch the partial sum approach ${\pi }^2/6$
• Exponent p: move to a $p$-series, and compare divergence for $p\le 1$ with convergence for $p>1$
• Partial sum / Area: switch between a running polyline and stacked areas
• Compare: place the harmonic series, the Basel series, and a geometric series in one frame, each scaled to its own range
• Euler: factors sit on a circle, and the picture writes the sine product whose $x^2$ coefficient this page identifies with $\sum 1/n^2={\pi }^2/6$
• p-series: when $p\le 1$ the frame marks a divergent region; when $p>1$ the partial sum approaches a finite value
• Zeros: the zeros of $\sin (x)/x$ at $\pm n\pi$, with the partial sum set beside ${\pi }^2/6$
• Pause / Play: stop or resume the reveal
• Replay: run the reveal again from the start

What to notice

• In partial-sum mode the polyline climbs toward ${\pi }^2/6$, and the error falls as $N$ grows.
• Compare mode draws the harmonic series, the Basel series, and a geometric series in one frame, each on its own scale. How fast they settle shows up in the shape of the curve.
• In p-series mode, $p\le 1$ marks a divergent region, and $p>1$ sends the partial sum toward a finite value.

Related

• Arithmetic and geometric sequences
• Fibonacci spiral
• Riemann sum

Further reading

• Basel problem (Wikipedia)