Basel Problem/Proof 2

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Theorem

$\displaystyle \map \zeta 2 = \sum_{n \mathop = 1}^{\infty} {\frac 1 {n^2}} = \frac {\pi^2} 6$

where $\zeta$ denotes the Riemann zeta function.


Proof


Let:

$\displaystyle P_k = x \prod_{n \mathop = 1}^k \paren {1 - \frac {x^2} {n^2 \pi^2} }$

We note that:

\(\displaystyle P_k - P_{k - 1}\) \(=\) \(\displaystyle \paren {-\frac {x^3} {k^2 \pi^2} } \prod_{n \mathop = 1}^{k - 1} \paren {1 - \frac {x^2} {n^2 \pi^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle -\frac {x^3} {k^2 \pi ^2} + \map O {x^5}\) Big-O Notation

Examples:

\(\displaystyle P_2 - P_1\) \(=\) \(\displaystyle x \paren {1 - \frac {x^2} {\pi^2} } \paren {\paren {1 - \frac {x^2} {4 \pi^2} } - 1 }\)
\(\displaystyle \) \(=\) \(\displaystyle -\frac {x^3} {4 \pi ^2} + \frac {x^5} {4 \pi ^4}\)


\(\displaystyle P_3 - P_2\) \(=\) \(\displaystyle x \paren {1 - \frac {x^2} {\pi^2} } \paren {1 - \frac {x^2} {4 \pi^2} } \paren {\paren {1 - \frac {x^2} {9 \pi^2} } - 1 }\)
\(\displaystyle \) \(=\) \(\displaystyle -\frac {x^3} {9 \pi ^2} + \frac {5 x^5} {36 \pi ^4} - \frac {x^7} {36 \pi ^4}\)


By Telescoping Series we find that the coefficient of $x^3$ in $P_k$ is given by:

$(1): \quad \displaystyle -\frac 1 {\pi^2} \sum_{i \mathop = 1}^k \frac 1 {i^2}$

We also have the following two representations of the Sine of x:

\(\displaystyle \sin x\) \(=\) \(\displaystyle x \prod_{n \mathop = 1}^\infty \paren {1 - \dfrac {x^2} {n^2 \pi^2} }\) Euler Formula for Sine Function
\(\displaystyle \sin x\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \dfrac {x^3} {3!} + \dfrac {x^5} {5!} - \dfrac {x^7} {7!} + \cdots\) Power Series Expansion for Sine Function

So by taking the limit as $k \to \infty$ in $(1)$ and equating with the coefficient of $x^3$ in the Power Series Expansion for Sine Function, we can deduce:

$\displaystyle -\frac 1 {\pi^2} \sum_{i \mathop = 1}^{\infty} \frac 1 {i^2} = -\frac 1 {3!}$

hence:

$\displaystyle \sum_{i \mathop = 1}^{\infty} \frac 1 {i^2} = \frac {\pi^2} 6$

$\blacksquare$


Historical Note

The Basel Problem was first posed by Pietro Mengoli in $1644$.

Its solution is generally attributed to Leonhard Euler , who solved it in $1734$ and delivered a proof in $1735$.

However, it has also been suggested that it was in fact first solved by Nicolaus I Bernoulli.

Jacob Bernoulli had earlier established that the series was convergent, but had failed to work out what to.

The problem is named after Basel, the home town of Euler as well as of the Bernoulli family.


If only my brother were alive now.
-- Johann Bernoulli


Sources