Sum of Reciprocals of Squares of Odd Integers/Proof 4
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Theorem
\(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}\) | \(=\) | \(\ds 1 + \dfrac 1 {3^2} + \dfrac 1 {5^2} + \dfrac 1 {7^2} + \dfrac 1 {9^2} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\pi^2} 8\) |
Proof
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\(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2n - 1}^2}\) | \(=\) | \(\ds \int_0^1 \int_0^1 \frac 1 {1 - x^2y^2} \rd x \rd y\) | Sum of Reciprocals of Squares of Odd Integers as Double Integral |
Applying the substitution:
- $\tuple {x, y} = \tuple {\dfrac {\sin u} {\cos v}, \dfrac {\sin v} {\cos u} }$
the Jacobian matrix is:
- $\mathbf J_{\mathbf f} := \begin{pmatrix} {\dfrac \partial {\partial u} \dfrac {\sin u} {\cos v} } & {\dfrac \partial {\partial v} \dfrac {\sin u} {\cos v} } \\ {\dfrac \partial {\partial u} \dfrac {\sin v} {\cos u} } & {\dfrac \partial {\partial v} \dfrac {\sin v} {\cos u} } \end{pmatrix}$
and the Jacobian determinant is:
\(\ds \size {\frac {\partial \tuple {x, y} } {\partial \tuple {u, v} } }\) | \(=\) | \(\ds \frac \partial {\partial u} \frac {\sin u} {\cos v} \frac \partial {\partial v} \frac {\sin v} {\cos u} - \frac \partial {\partial v} \frac {\sin u} {\cos v} \frac \partial {\partial u} \frac {\sin v} {\cos u}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cos u \, \cos v} {\cos v \, \cos u} - \frac {\sin^2 u \, \sin^2 v} {\cos^2 v \, \cos^2 u}\) | Derivative of Secant Function, Derivative of Sine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \tan^2 u \tan^2 v\) |
Under this substitution, the image of the region $\closedint 0 1^2$, that is the unit square, is an isosceles triangle $\bigtriangleup$ with:
- base and height $\dfrac \pi 2$
- vertices: $\tuple {0, 0}; \tuple {0, \dfrac \pi 2}; \tuple {\dfrac \pi 2, 0}$
We have:
\(\ds 0\) | \(\le\) | \(\ds \dfrac {\sin u} {\cos v}\) | \(\ds \le 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds u\) | \(\ge\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin u\) | \(\le\) | \(\ds \cos v\) |
and we have:
\(\ds 0\) | \(\le\) | \(\ds \dfrac {\sin v} {\cos u}\) | \(\ds \le 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(\ge\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin v\) | \(\le\) | \(\ds \cos u\) |
This gives us the region:
- $\set {\tuple {u, v}: u, v \ge 0 \land \cos u \ge \sin v \land \cos v \ge \sin u}$
which is equivalent to the region:
- $\set {\tuple {u, v}: u, v \ge 0 \land v \le \dfrac \pi 2 - u}$
By Change of Variables Theorem (Multivariable Calculus):
\(\ds \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y\) | \(=\) | \(\ds \iint_{\bigtriangleup} \frac {1 - \tan^2 u \, \tan^2 v} {1 - \paren {\tan u \, \tan v}^2} \rd u \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \iint_{\bigtriangleup} \rd u \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^{\frac \pi 2} \int_0^{\frac \pi 2} \rd u \rd v\) | Area of Triangle in Terms of Side and Altitude | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac \pi 2}^2\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^2} 8\) |
$\blacksquare$