Linked Questions

2
votes
2answers
644 views

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 $0≤x_1 ≤y_1$ ...
3
votes
2answers
217 views

Question about arithmetic–geometric mean [duplicate]

We have two sequences: $$a_{n+1}=\sqrt{a_nb_n}$$ $$b_{n+1}=\frac{a_n+b_n}{2}$$ I need to prove that those are making Cantor's Lemma.(At the end I shold get that: $\lim_{n\to \infty}a_n=\lim_{n\to \...
2
votes
2answers
98 views

for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}$ [duplicate]

for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}, b_{n+1} = \frac{a_n + b_n}{2}$. Show that $a_n, b_n$ converge to a common limit. Hint use inequality: $...
0
votes
3answers
93 views

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 ...
3
votes
1answer
54 views

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 ...
0
votes
1answer
49 views

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 $b_1\le a_1$ ...
1
vote
0answers
62 views

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 ...
0
votes
0answers
45 views

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 ...
7
votes
3answers
3k views

Proof that arithmetic and geometric mean converge

I need some help with understanding a part of this proof and also writing it up correctly. Given $a_n\geq a_{n+1}\geq b_{n+1} \geq b_n$ with $a_1=a$ and $b_1=b$. I am also given that $$a_{n+1}=\frac{...
4
votes
1answer
1k views

These two sequences have the same limit

Let $a_1$ and $b_1$ be any two positive numbers, and define $\{ a_n\}$ and $\{ b_n\}$ by $$a_n = \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}},$$ $$b_n = \sqrt{a_{n-1}b_{n-1} }.$$ Prove that the ...
2
votes
2answers
164 views

Arithmetic-geometric mean convergence proof.

So in my book A friendly introduction to Analysis, there's an exercise that I'm having trouble with. The exercise is as follow: Consider sequences {$a_n$} and {$b_n$} which satisfy: $0 < b_1 < ...
1
vote
2answers
103 views

Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$

If $a$ and $b$ are positive real numbers such that $a<b$ and if $$a_1=\frac{a+b}{2}, b_1=\sqrt{(a_1b)},..., a_n=\frac{a_{n-1}+b_{n-1}}{2},b_n=\sqrt{a_nb_{n-1}},$$ then show that $$\lim_{n\to\...
2
votes
1answer
202 views

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 ...
3
votes
1answer
297 views

Real Analysis Arithmetic-Geometric Mean Question

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}{...
2
votes
3answers
98 views

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),(...

15 30 50 per page