Sum of Sequence of Reciprocals of 3 n + 2 Alternating in Sign

From ProofWiki
Jump to navigation Jump to search

Theorem

\(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {3 n + 2}\) \(=\) \(\displaystyle \frac 1 2 - \frac 1 5 + \frac 1 8 - \frac 1 {11} + \frac 1 {14} - \cdots\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\pi \sqrt 3} 9 - \dfrac {\ln 2} 3\)


Proof

\(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {3 n + 2}\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^1 x^{3 n + 1} \rd x\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle \int_0^1 x \paren {\sum_{n \mathop = 0}^\infty \paren {-x^3}^n} \rd x\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle \int_0^1 \frac x {x^3 + 1} \rd x\) Sum of Infinite Geometric Progression
\(\displaystyle \) \(=\) \(\displaystyle \intlimits {\frac 1 6 \ln \size {\frac {x^2 - x + 1} {\paren {x + 1}^2} } + \frac 1 {\sqrt 3} \map \arctan {\frac {2 x - 1} {\sqrt 3} } } 0 1\) Primitive of $\dfrac x {x^3 + a^3}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 6 \map \ln {\frac 1 4} + \frac 1 {\sqrt 3} \map \arctan {\frac 1 {\sqrt 3} } - \frac 1 6 \ln 1 - \frac 1 {\sqrt 3} \map \arctan {-\frac 1 {\sqrt 3} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 {\sqrt 3} \map \arctan {\frac 1 {\sqrt 3} } -\frac 1 3 \ln 2\) Logarithm of Power, Natural Logarithm of 1 is 0, Arctangent Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 {\sqrt 3} \times \frac \pi 6 - \frac 1 3 \ln 3\) Arctangent of $\dfrac {\sqrt 3} 3$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\pi \sqrt 3} 9 - \frac {\ln 2} 3\)

$\blacksquare$


Sources