Editing Pi (section)

==== Rate of convergence ====
Some infinite series for {{pi}} [[convergent series|converge]] faster than others. Given the choice of two infinite series for {{pi}}, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate {{pi}} to any given accuracy.<ref name="Aconverge">{{cite journal |last1=Borwein |first1=J. M. |last2=Borwein |first2=P. B. |title=Ramanujan and Pi |year=1988 |journal=Scientific American |volume=256 |issue=2 |pages=112–117 |bibcode=1988SciAm.258b.112B |doi=10.1038/scientificamerican0288-112}}{{br}} {{harvnb|Arndt|Haenel|2006|pp=15–17, 70–72, 104, 156, 192–197, 201–202}}.</ref> A simple infinite series for {{pi}} is the [[Leibniz formula for π|Gregory–Leibniz series]]:{{sfn|Arndt|Haenel|2006|pp=69–72}}

<math display=block>
\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots
</math>

As individual terms of this infinite series are added to the sum, the total gradually gets closer to {{pi}}, and&nbsp;– with a sufficient number of terms&nbsp;– can get as close to {{pi}} as desired. It converges quite slowly, though&nbsp;– after 500,000 terms, it produces only five correct decimal digits of {{pi}}.<ref>{{cite journal |last1=Borwein |first1=J. M. |last2=Borwein |first2=P. B. |last3=Dilcher |first3=K. |year=1989 |title=Pi, Euler Numbers, and Asymptotic Expansions |journal=American Mathematical Monthly |volume=96 |issue=8 |pages=681–687 |doi=10.2307/2324715 |jstor=2324715 |hdl=1959.13/1043679 |hdl-access=free}}</ref>

An infinite series for {{pi}} (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:{{sfn|Arndt|Haenel|2006|loc = Formula 16.10, p. 223}}<ref>{{cite book |last=Wells |first=David |page=35 |title=The Penguin Dictionary of Curious and Interesting Numbers |edition=revised |publisher=Penguin |year=1997 |isbn=978-0-14-026149-3}}</ref>

<math display=block>
\pi = 3 + \frac{4}{2\times3\times4} - \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8} - \frac{4}{8\times9\times10} + \cdots
</math>

The following table compares the convergence rates of these two series:

{|class="wikitable" style="text-align: center; margin: auto;"
|-
! Infinite series for {{pi}} !! After 1st term !! After 2nd term !! After 3rd term !! After 4th term !! After 5th term !! Converges to:
|-
| <math>\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} + \cdots</math>
||4.0000||2.6666 ... ||3.4666 ... ||2.8952 ... ||3.3396 ... ||rowspan=2| {{pi}} = 3.1415 ...
|-
| <math>\pi = {{3}} + \frac{{4}}{2\times3\times4} - \frac{{4}}{4\times5\times6} + \frac{{4}}{6\times7\times8} - \cdots </math>
||3.0000||3.1666 ... ||3.1333 ... ||3.1452 ... ||3.1396 ...
|}

After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of {{pi}}, whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of {{pi}}. Series that converge even faster include [[Machin-like formula|Machin's series]] and [[Chudnovsky algorithm|Chudnovsky's series]], the latter producing 14 correct decimal digits per term.{{r|Aconverge}}