Power Series Expansion for Real Arctangent Function
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Theorem
The arctangent function has a Taylor series expansion:
- $\arctan x = \begin {cases} \ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \ds -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \le -1 \end {cases}$
That is:
- $\arctan x = \begin {cases} x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 - \cdots & : -1 \le x \le 1 \\ \dfrac \pi 2 - \dfrac 1 x + \dfrac 1 {3 x^3} - \dfrac 1 {5 x^5} + \cdots & : x \ge 1 \\ -\dfrac \pi 2 - \dfrac 1 x + \dfrac 1 {3 x^3} - \dfrac 1 {5 x^5} + \cdots & : x \le -1 \end {cases}$
Proof
From Sum of Infinite Geometric Sequence:
- $(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x^2}^n = \frac 1 {1 + x^2}$
for $-1 < x < 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
\(\ds \int_0^x \frac 1 {1 + t^2} \rd t\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \int_0^x \paren {-t^2}^n \rd t\) | ||||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \arctan x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1}\) | Primitive of Reciprocal of $\dfrac 1 {1 + t^2}$, Integral of Power |
For $-1 \le x \le 1$, the sequence $\sequence {\dfrac {x^{2 n + 1}} {2 n + 1} }$ is decreasing and converges to zero.
Therefore the series converges in the given range by the Alternating Series Test.
$\Box$
Now consider the case $x \ge 1$:
\(\ds \arctan x\) | \(=\) | \(\ds \frac \pi 2 - \map \arccot x\) | Sum of Arctangent and Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2 - \map \arctan {\frac 1 x}\) | Arctangent of Reciprocal equals Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {2 n + 1} \paren {\frac 1 x}^{2 n + 1}\) | as $x \ge 1$, $0 < \dfrac 1 x \le 1$, so $(2)$ may be applied | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} }\) |
$\Box$
We also have:
\(\ds \map \arctan {-x}\) | \(=\) | \(\ds -\arctan x\) | Arctangent Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^{n + 1} \frac 1 {\paren {2 n + 1} x^{2 n + 1} }\) | from $(3)$ |
Substituting $x$ for $-x$ gives us the expansion for $x \le -1$:
\(\ds \arctan x\) | \(=\) | \(\ds -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^{n + 1} \frac 1 {\paren {2 n + 1} \paren {-x}^{2 n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^{n + 1} \frac 1 {-\paren {2 n + 1} x^{2 n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} }\) |
$\blacksquare$
Also see
- Power Series Expansion for Real Arcsine Function
- Power Series Expansion for Real Arccosine Function
- Power Series Expansion for Real Arccotangent Function
- Power Series Expansion for Real Arcsecant Function
- Power Series Expansion for Real Arccosecant Function
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.29$
- 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