Political Science Observations

The Basel Problem

One of the most famous problems in mathematical history

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}\]

What is the Basel Problem?

The Basel problem is a question about the exact value of the infinite series:

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots\]

First posed in 1650 by Pietro Mengoli and later popularized by the Bernoulli family, the problem remained unsolved for nearly a century until the brilliant mathematician Leonhard Euler found the solution in 1734.

The problem is named after Basel, Switzerland, the hometown of both Euler and the Bernoulli family.

Euler's Ingenious Solution

Euler's solution was remarkable for its creativity and boldness. Here's an overview of his approach:

Sine Function Expansion

Euler began with the Taylor series expansion of the sine function:

\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]

Alternative Representation

He then considered the sine function as an infinite product:

\[\sin x = x \left(1 - \frac{x^2}{\pi^2}\right) \left(1 - \frac{x^2}{4\pi^2}\right) \left(1 - \frac{x^2}{9\pi^2}\right) \cdots\]

Comparing Coefficients

By expanding the product and comparing the coefficients of \(x^3\) from both representations, Euler found:

\[-\frac{1}{3!} = -\left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots\right)\]

Final Solution

Multiplying both sides by \(-\pi^2\) gives:

\[\frac{\pi^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots\]

Euler's approach was not initially rigorous by modern standards, as he treated infinite series somewhat informally. However, his result was correct, and his methods were later justified with more rigorous approaches.

Significance of the Solution

Euler's solution to the Basel problem was a landmark achievement in mathematics:

Unexpected Connection

It revealed a surprising connection between number theory (the sum of reciprocals of squares) and geometry (π).

Zeta Function

It established the importance of the Riemann zeta function, defined as \(\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\), with \(\zeta(2) = \frac{\pi^2}{6}\).

Infinite Series

It advanced the theory of infinite series and inspired further research in mathematical analysis.

Euler's Reputation

It solidified Euler's reputation as one of the greatest mathematicians of all time.

Convergence of the Series

The Basel series converges relatively quickly compared to the harmonic series. Here's how the partial sums approach the limit:

Let \(S_N = \sum_{n=1}^{N} \frac{1}{n^2}\) be the partial sum after N terms.

Then \(\lim_{N \to \infty} S_N = \frac{\pi^2}{6} \approx 1.6449340668482264\)

Partial Sums Convergence

As we add more terms, the partial sums get closer to π²/6:

\(S_1 = 1\)
\(S_2 = 1 + \frac{1}{4} = 1.25\)
\(S_3 = 1 + \frac{1}{4} + \frac{1}{9} \approx 1.3611\)
\(S_5 \approx 1.4636\)
\(S_{10} \approx 1.5498\)
\(S_{50} \approx 1.6251\)
\(S_{100} \approx 1.6350\)
\(S_{1000} \approx 1.6439\)

Extensions and Related Results

Euler's solution opened the door to finding values of the zeta function at other even integers:

\(\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}\)
\(\zeta(6) = \sum_{n=1}^{\infty} \frac{1}{n^6} = \frac{\pi^6}{945}\)
\(\zeta(8) = \sum_{n=1}^{\infty} \frac{1}{n^8} = \frac{\pi^8}{9450}\)

In general, Euler proved that for positive integers k:

\[\zeta(2k) = (-1)^{k+1} \frac{B_{2k} (2\pi)^{2k}}{2(2k)!}\]

where \(B_{2k}\) are Bernoulli numbers.

Interestingly, the values of ζ(s) at odd integers are much more mysterious and no simple closed-form expressions are known for them.