Editing Fibonacci sequence (section)

=== Computation by rounding ===
Since
<math display=inline>\left|\frac{\psi^{n}}{\sqrt 5}\right| < \frac{1}{2}</math> for all {{math|''n'' ≥ 0}}, the number {{math|''F''<sub>''n''</sub>}} is the closest [[integer]] to <math>\frac{\varphi^n}{\sqrt 5}</math>. Therefore, it can be found by [[rounding]], using the nearest integer function:
<math display=block>F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5}\right\rceil,\ n \geq 0.</math>

In fact, the rounding error quickly becomes very small as {{mvar|n}} grows, being less than 0.1 for {{math|''n'' ≥ 4}}, and less than 0.01 for {{math|''n'' ≥ 8}}.  This formula is easily inverted to find an index of a Fibonacci number {{mvar|F}}:
<math display=block>n(F) = \left\lfloor  \log_\varphi \sqrt{5}F\right\rceil,\ F \geq 1.</math>

Instead using the [[floor function]] gives the largest index of a Fibonacci number that is not greater than {{mvar|F}}:
<math display=block>n_{\mathrm{largest}}(F) = \left\lfloor  \log_\varphi \sqrt{5}(F+1/2)\right\rfloor,\ F \geq 0,</math>
where <math>\log_\varphi(x) = \ln(x)/\ln(\varphi) = \log_{10}(x)/\log_{10}(\varphi)</math>, <math>\ln(\varphi) = 0.481211\ldots</math>,<ref>{{Cite OEIS|1=A002390|2=Decimal expansion of natural logarithm of golden ratio|mode=cs2}}</ref> and <math>\log_{10}(\varphi) = 0.208987\ldots</math>.<ref>{{Cite OEIS|1=A097348|2=Decimal expansion of arccsch(2)/log(10)|mode=cs2}}</ref>