# AGM Calculator

## Calculates the arithmetic-geometric mean (AGM).

 $Arithmetic-geometric\ mean\ AGM(a,b)\\[15]a_0=a,\ b_0=b\\\vspace{20}a_{n+1}={\large\frac{1}{2}}(a_n+b_n),\hspace{20}b_{n+1}=\sqrt{a_nb_n},\\\vspace{20}AGM(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n\\$
 a b [ maximum loop n ]
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 The calculation ends when two consecutive results are the same.The accuracy of AGM improves by increasing the number of digits for calculation. $\normal Arithmetic-geometric\ mean\ AGM(a,b)\\[10](1)\ a_0=a,\ b_0=b\\\vspace{20}(2)\ a_{n+1}={\large\frac{1}{2}}(a_n+b_n),\hspace{20}b_{n+1}=\sqrt{a_nb_n},\\\vspace{20}(3)\ a_1\ge a_2\ge a_3\ge ... \ge b_3\ge b_2\ge b_1\\\vspace{20}(4)\ AGM(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n\\$

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