Editing Logarithm (section)

====Taylor series====

[[File:Taylor approximation of natural logarithm.gif|right|thumb|The Taylor series of {{math|ln(''z'')}} centered at {{math|''z'' {{=}} 1}}. The animation shows the first&nbsp;10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.|alt=An animation showing increasingly good approximations of the logarithm graph.]]
For any real number {{mvar|z}} that satisfies {{math|0 < ''z'' ≤ 2}}, the following formula holds:{{refn|The same series holds for the principal value of the complex logarithm for complex numbers {{mvar|z}} satisfying {{math|{{!}}''z'' − 1{{!}} < 1}}.|group=nb}}<ref name=AbramowitzStegunp.68>{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972 |nb=yes|loc=p. 68}}</ref>

<math display="block">
\begin{align}\ln (z)  &= \frac{(z-1)^1}{1} - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \cdots \\
&= \sum_{k=1}^\infty (-1)^{k+1}\frac{(z-1)^k}{k}.
\end{align}
</math>

Equating the function {{math|ln(''z'')}} to this infinite sum ([[series (mathematics)|series]]) is shorthand for saying that the function can be approximated to a more and more accurate value by the following expressions (known as [[partial sum]]s):

<math display=block>
(z-1),\ \ (z-1) - \frac{(z-1)^2}{2},\ \ (z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3},\ \ldots
</math>

For example, with {{math|''z'' {{=}} 1.5}} the third approximation yields {{math|0.4167}}, which is about {{math|0.011}} greater than {{math|ln(1.5) {{=}} 0.405465}}, and the ninth approximation yields {{math|0.40553}}, which is only about {{math|0.0001}} greater. The {{mvar|n}}th partial sum can approximate {{math|ln(''z'')}} with arbitrary precision, provided the number of summands {{mvar|n}} is large enough.

In elementary calculus, the series is said to [[convergent series|converge]] to the function {{math|ln(''z'')}}, and the function is the [[limit (mathematics)|limit]] of the series. It is the [[Taylor series]] of the [[natural logarithm]] at {{math|1=''z'' = 1}}. The Taylor series of {{math|ln(''z'')}} provides a particularly useful approximation to {{math|ln(1 + ''z'')}} when {{mvar|z}} is small, {{math|{{!}}''z''{{!}} < 1}}, since then
<math display="block">
\ln (1+z) = z - \frac{z^2}{2}  +\frac{z^3}{3} -\cdots \approx z.
</math>

For example, with {{math|1=''z'' = 0.1}} the first-order approximation gives {{math|ln(1.1) ≈ 0.1}}, which is less than {{math|5%}} off the correct value {{math|0.0953}}.