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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 $$\int_ {0}^{\pi/2}\frac{1}{\sqrt{a^2\cos^2t+b^2\sin^2t}}dt=\frac{\pi}{2\mathrm{AGM}(a,b)}.$$

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