Sum of Reciprocals of Squares of Odd Integers/Proof 3
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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
This "proof" leads to a dead end. Its value lies in showing that this approach does not work.
\(\ds \) | \(\) | \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y\) | Sum of Reciprocals of Squares of Odd Integers as Double Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \int_0^1 \frac 1 {\paren {1 + x y} \paren {1 - x y} } \rd x \rd y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \int_0^1 \frac 1 {2 \paren {1 + x y} } \rd x \rd y - \int_0^1 \int_0^1 \frac 1 {2 \paren {1 - x y} } \rd x \rd y\) | Primitive of Function of Constant Multiple, partial fraction expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^1 \intlimits {\frac {\map \ln {1 + x y} } x} 0 1 \rd x - \frac 1 2 \int_0^1 \intlimits {\frac {\map \ln {1 - x y} } x} 0 1 \rd x\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^1 \paren {\frac {\map \ln {1 + x} } x - \frac {\ln 1} x} \rd x - \frac 1 2 \int_0^1 \paren {\frac {\map \ln {1 - x} } x - \frac {\ln 1} x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^1 \frac 1 x \map \ln {\frac {1 + x} {1 - x} } \rd x\) | Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \frac {\artanh x} x \rd x\) | Definition of Real Area Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \frac 1 x \paren {x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots} \rd x\) | Power Series Expansion for Real Area Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \paren {1 + \frac {x^2} 3 + \frac {x^4} 5 + \frac {x^6} 7 + \cdots} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {x + \frac {x^3} 9 + \frac {x^5} {25} + \frac {x^7} {49} + \cdots} 0 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \frac 1 9 + \frac 1 {25} + \frac 1 {49} + \cdots\) | and a dead end is reached |