Editing Lorentz factor (section)

===Series expansion (velocity)===
The Lorentz factor has the [[Taylor series|Maclaurin series]]:
<math display="block">\begin{align}
\gamma & = \dfrac{1}{\sqrt{1 - \beta^2}} \\[1ex]
& = \sum_{n=0}^{\infty} \beta^{2n}\prod_{k=1}^n \left(\dfrac{2k - 1}{2k}\right) \\[1ex]
& = 1 + \tfrac12 \beta^2 + \tfrac38 \beta^4 + \tfrac{5}{16} \beta^6 + \tfrac{35}{128} \beta^8 + \tfrac{63}{256} \beta^{10} + \cdots ,
\end{align}</math>
which is a special case of a [[binomial series]].

The approximation <math display="inline">\gamma \approx 1 + \frac{1}{2}\beta^2</math> may be used to calculate relativistic effects at low speeds. It holds to within 1% error for {{mvar|v}}&nbsp;&lt; 0.4&nbsp;{{mvar|c}} ({{mvar|v}}&nbsp;&lt; 120,000&nbsp;km/s), and to within 0.1% error for {{mvar|v}}&nbsp;&lt; 0.22&nbsp;{{mvar|c}} ({{mvar|v}}&nbsp;&lt; 66,000&nbsp;km/s).

The truncated versions of this series also allow [[physicist]]s to prove that [[special relativity]] reduces to [[Newtonian mechanics]] at low speeds. For example, in special relativity, the following two equations hold:

<math display="block">\begin{align}
\mathbf p & = \gamma m \mathbf v, \\
E & = \gamma m c^2.
\end{align}</math>

For <math>\gamma \approx 1</math> and <math display="inline">\gamma \approx 1 + \frac{1}{2}\beta^2</math>, respectively, these reduce to their Newtonian equivalents:

<math display="block">\begin{align}
\mathbf p & = m \mathbf v, \\
E & = m c^2 + \tfrac12 m v^2.
\end{align}</math>

The Lorentz factor equation can also be inverted to yield
<math display="block">\beta = \sqrt{1 - \frac{1}{\gamma^2}} .</math>
This has an asymptotic form
<math display="block">\beta = 1 - \tfrac12 \gamma^{-2} - \tfrac18 \gamma^{-4} - \tfrac{1}{16} \gamma^{-6} - \tfrac{5}{128} \gamma^{-8} + \cdots\,.</math>

The first two terms are occasionally used to quickly calculate velocities from large {{mvar|γ}} values. The approximation <math display="inline">\beta \approx 1 - \frac{1}{2}\gamma^{-2}</math> holds to within 1% tolerance for {{nobr|{{math|''γ'' > 2}},}} and to within 0.1% tolerance for {{nobr|{{math|''γ'' > 3.5}}.}}