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### Arithmetic-geometric Mean [duplicate]

(Arithmetic–Geometric Mean). (a) Explain why $\sqrt{xy}≤ (x+y)/2$ for any two positive real numbers x and y. (The geometric mean is always less than the arithmetic mean.) (b) Now let $0≤x_1 ≤y_1$ ...
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### Prove that two sequences converge to the same limit. [duplicate]

I encountered this question in my homework: $$a_1=x, b_1=y, \\ a_{n+1} =\frac{a_n+b_n}{2}, b_{n+1}= \sqrt{a_nb_n}, n\in \mathbb{N}$$ Given $x,y$ positive constants. I have to prove that they both ...
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### Show that both sequences converge to a common limit [duplicate]

Show that if $a_1>b_1>0,a_{n+1}=\sqrt{a_nb_n}$ and $b_{n+1}=\frac{a_n+b_n}{2}$, then $a_n$ and $b_n$ both converge to a common limit. It seems a bit intuitive but I am not able to get it in ...
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### Series composed of arithmetic and geometric mean properties [duplicate]

We have that $0<b\le a$ are two positive real numbers. Their arithmetic mean is defined as $a_1=(a+b)/2$ and their geometric mean is $b_1=\sqrt{ab}$. Firstly we have to show an that $b_1\le a_1$ ...
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### Prove $a_n$ and $b_n$ have the same limit [duplicate]

Let x,y be positive, constant nambers so: $a_1=x, b_1=y$ and $a_{n+1}=(a_n+b_n)/2,\; b_{n+1}=\sqrt{a_nb_n}$ Prove that $a_n$ and $b_n$ have the same limit. From inequality of arithmetic and ...
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### Monotone Convergence Theorem Question [duplicate]

I'm having trouble with the following problem: Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences satisfying $$a_{n + 1} = \frac{a_{n} + b_{n}}{2}$$ and $$b_{n + 1} = \sqrt{a_{n}b_{n}}$$ Show that ...
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### Prove that $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have the same limit

I'm trying to solve the following problem prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit. In this post https://math.stackexchange.com/a/267499, I do not understand the following ...
I am working on a multiple part question for an introductory Real Analysis course. I have part of it done, but I have some problems. Let $0 < y_1 < x_1$, and set x_{n+1}=\frac{x_n +y_n}{...
### How can we show that $(a_n), (b_n), (c_n)$ are convergent and have the same limit?
We have the following for $a \le b \le c >0$: $A(a,b,c)=\frac{a+b+c}{3}, B(a,b,c)= (abc)^{1/3}, C(a,b,c)=\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$. Then we define the sequences \$(a_n),(...