Basel Problem/Proof 9: Difference between revisions
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== Sources == | == Sources == | ||
* {{BookReference|Fourier Series|1961|I.N. Sneddon|prev = Fourier Series/Square of x minus pi, Square of pi|next = Sum of Reciprocals of Squares Alternating in Sign/Proof 1}}: Chapter One: $\S 2$. Fourier Series: Example $1$ | * {{BookReference|Fourier Series|1961|I.N. Sneddon|prev = Fourier Series/Square of x minus pi, Square of pi/Mistake|next = Sum of Reciprocals of Squares Alternating in Sign/Proof 1}}: Chapter One: $\S 2$. Fourier Series: Example $1$ | ||
[[Category:Basel Problem]] | [[Category:Basel Problem]] |
Revision as of 07:32, 7 March 2018
Theorem
- $\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function.
Proof
Let $f \left({x}\right)$ be the real function defined on $\left({0 \,.\,.\, 2 \pi}\right)$ as:
- $f \left({x}\right) = \begin{cases}
\left({x - \pi}\right)^2 & : 0 < x \le \pi \\ \pi^2 & : \pi < x < 2 \pi \end{cases}$
From Fourier Series: Square of x minus pi, Square of pi, its Fourier series can be expressed as:
- $f \left({x}\right) \sim \displaystyle \frac {2 \pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\frac {2 \cos n x} {n^2} + \left({\frac {\left({-1}\right)^n \pi} n + \frac {2 \left({\left({-1}\right)^n - 1}\right)} {\pi n^3} }\right) \sin n x}\right)$
Setting $x = 0$:
\(\ds f \left({0}\right)\) | \(=\) | \(\ds \frac {2 \pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\frac {2 \cos 0} {n^2} + \left({\frac {\left({-1}\right)^n \pi} n + \frac {2 \left({\left({-1}\right)^n - 1}\right)} {\pi n^3} }\right) \sin 0}\right)\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \left({0 - \pi}\right)^2\) | \(=\) | \(\ds \frac {2 \pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\frac {2 \cos 0} {n^2} }\right)\) | Sine of Zero is Zero | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pi^2\) | \(=\) | \(\ds \frac {2 \pi^2} 3 + 2 \sum_{n \mathop = 1}^\infty \frac 1 {n^2}\) | Cosine of Zero is One | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\pi^2} 2 - \frac {\pi^2} 3\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\pi^2} 6\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^2}\) |
$\blacksquare$
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 2$. Fourier Series: Example $1$