Power Series Expansion for Cotangent Function
Theorem
The (real) cotangent function has a Taylor series expansion:
| \(\ds \cot x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \frac {x^7} {4725} - \cdots\) |
where $B_{2 n}$ denotes the Bernoulli numbers.
This converges for $0 < \size x < \pi$.
Proof
| \(\ds \cot x\) | \(=\) | \(\ds i \frac {e^{i x} + e^{- i x} } {e^{i x} - e^{- i x} }\) | Euler's Cotangent Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds i \frac {e^{2 i x} + 1 } {e^{2 i x} - 1 }\) | multiplying top and bottom by $e^{ix}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds i \paren {\frac {e^{2 i x} - 1 + 2 } {e^{2 i x} - 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i \paren {1 + \frac 2 {e^{2 i x} - 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i + \frac {2 i} {e^{2 i x} - 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i + \frac 1 x \frac {2 i x} {e^{2 i x} - 1}\) | multiplying top and bottom by $x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds i + \frac 1 x \sum_{n \mathop = 0}^\infty \frac {B_n \paren {2 i x}^n} {n!}\) | Definition of Bernoulli Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds i + \frac 1 x \paren { 1 + \frac {-\dfrac 1 2 \paren {2 i x} } {1!} + \sum_{n \mathop = 2}^\infty \frac {B_n \paren {2 i x}^n} {n!} }\) | as $B_0 = 1$ and $B_1 = -\dfrac 1 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 x + \frac 1 x \sum_{n \mathop = 2}^\infty \frac {B_n \paren {2 i x}^n} {n!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 x + \frac 1 x \sum_{n \mathop = 1}^\infty \frac {B_{2 n} \paren {2 i x}^{2 n} } {\paren {2 n}!}\) | Odd Bernoulli Numbers Vanish | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 x \sum_{n \mathop = 0}^\infty \frac {B_{2 n} \paren {2 i x}^{2 n} } {\paren {2 n}!}\) | as $B_0 = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n} } {\paren {2 n}!}\) | $i^2 = -1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}\) | Moving $\dfrac 1 x $ inside the sum |
By Combination Theorem for Limits of Real Functions we can deduce the following.
| \(\ds \) | \(\) | \(\ds \lim_{n \mathop \to \infty} \size {\frac {\frac {\paren {-1}^n 2^{2 n + 2} B_{2 n + 2} \, x^{2 n + 1} } {\paren {2 n + 2}!} } {\frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size {\frac 1 {\paren {2 n + 1} \paren {2 n + 2} } \frac {B_{2 n + 2} } {B_{2 n} } } 4 x^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size {\frac 1 {\paren {2 n + 1} \paren {n + 1} } \frac {B_{2 n + 2} } {B_{2 n} } } 2 x^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size{\frac 1 {\paren {2 n + 1} \paren {n + 1} } \frac {\paren {-1}^{n + 2} 4 \sqrt {\pi (n + 1)} \paren {\frac {n + 1} {\pi e} }^{2 n + 2} } {\paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\frac n {\pi e} }^{2 n} } } 2 x^2\) | Asymptotic Formula for Bernoulli Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size {\frac {\paren {n + 1}^2} {\paren {2 n + 1} \paren {n + 1} } \sqrt {\frac {n + 1} n } \paren {\frac {n + 1} n}^{2 n} } \frac 2 {\pi^2 e^2} x^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size {\paren {\frac {n + 1} n}^{2 n} } \frac {x^2} {\pi^2 e^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \size {\paren {\paren {1 + \frac 1 n}^n}^2} \frac {x^2} {\pi^2 e^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^2 x^2} {\pi^2 e^2}\) | Definition of Euler's Number as Limit of Sequence | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {x^2} {\pi^2}\) |
This is less than $1$ if and only if:
- $\size x < \pi$
Hence by the Ratio Test, the series converges for $\size x < \pi$.
$\blacksquare$
Also presented as
The can also be presented in the form:
| \(\ds \cot x\) | \(=\) | \(\ds \dfrac 1 x - \sum_{n \mathop = 1}^\infty \frac {2^{2 n} {B_n}^* x^{2 n - 1} } {\paren {2 n}!}\) |
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.24$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.24.$