# Some means and sequence limit

If two sequences $$\{a_n\},\{b_n\}$$ satisfy $$a_0=a,b_0=b(a>0,b>0)$$, and $$\begin{cases} a_{n+1}=\frac{a_n+b_n}{2}\\ b_{n+1}=\sqrt{\frac{a_n^2+b_n^2}{2}}. \end{cases}(n=0,1,2,\ldots)$$ Find the limit of $$\{a_n\},\{b_n\}$$.

How about $$\begin{cases} a_{n+1}=\frac{2a_nb_n}{a_n+b_n}\\ b_{n+1}=\sqrt{\frac{a_n^2+b_n^2}{2}}. \end{cases}(n=0,1,2,\ldots)$$

As we know, the Arithmetic-Geometric Mean is, if two sequences $$\{a_n\},\{b_n\}$$ satisfy $$a_0=a,b_0=b(a>0,b>0)$$, and $$a_ {n+1}=\frac{a_n+b_n}2,\qquad b_{n+1}=\sqrt{a_nb_n}.$$ Let $$\mathrm{AGM}(a,b)$$ be the limit of $$\{a_n\},\{b_n\}$$,Then $$\int_ {0}^{\pi/2}\frac{1}{\sqrt{a^2\cos^2t+b^2\sin^2t}}dt=\frac{\pi}{2\mathrm{AGM}(a,b)}.$$

• So... What do you want? Asking for various methods? – Bach Nov 10 '18 at 7:50
• @Philip Find the limit of first two problems, which are different from AGM. – Eufisky Nov 10 '18 at 18:57
• @YuriyS Find the limit of problem in yellow texts. – Eufisky Nov 10 '18 at 19:02
• – Yuriy S Nov 10 '18 at 22:59
• – Yuriy S Nov 10 '18 at 23:23