Editing Arithmetic–geometric mean (section)

==Example==
To find the arithmetic–geometric mean of {{math|''a''<sub>0</sub> {{=}} 24}} and {{math|''g''<sub>0</sub> {{=}} 6}}, iterate as follows:<math display=block>\begin{array}{rcccl}
 a_1 & = & \tfrac12(24 + 6) & = & 15\\
 g_1 & = & \sqrt{24 \cdot 6} & = & 12\\
 a_2 & = & \tfrac12(15 + 12) & = & 13.5\\
 g_2 & = & \sqrt{15 \cdot 12} & = & 13.416\ 407\ 8649\dots\\
 & & \vdots & &
\end{array}</math>The first five iterations give the following values:

{| class="wikitable plainrowheaders" style="margin-left:2em;"
|-
! scope="col" | {{math|''n''}}
! scope="col" | {{math|''a''<sub>''n''</sub>}}
! scope="col" | {{math|''g''<sub>''n''</sub>}}
|-
! scope="row" | 0
| 24
| 6
|-
! scope="row" | 1
| {{underline|1}}5
| {{underline|1}}2
|-
! scope="row" | 2
| {{underline|13}}.5
| {{underline|13}}.416 407 864 998 738 178 455 042...
|-
! scope="row" | 3
| {{underline|13.458}} 203 932 499 369 089 227 521...
| {{underline|13.458}} 139 030 990 984 877 207 090...
|-
! scope="row" | 4
| {{underline|13.458 171 481 7}}45 176 983 217 305...
| {{underline|13.458 171 481 7}}06 053 858 316 334...
|-
! scope="row" | 5
| {{underline|13.458 171 481 725 615 420 766 8}}20...
| {{underline|13.458 171 481 725 615 420 766 8}}06...
|}

The number of digits in which {{math|''a''<sub>''n''</sub>}} and {{math|''g''<sub>''n''</sub>}} agree (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately {{val|13.4581714817256154207668131569743992430538388544}}.<ref>[http://www.wolframalpha.com/input/?i=agm%2824%2C+6%29 agm(24, 6)] at [[Wolfram Alpha]]</ref>