0
votes
1answer
15 views

Prove boundedness of recurrence relation

For a number sequence $\{y_n\}$ we know that $y_{n+1} = 2y_n-y^2_n$ If: $0<y_0<1$ show that $0<y_n<1$ for all integers $n>0$ I've tried solving the recurrence relation, but I couldn't ...
1
vote
1answer
48 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
5
votes
3answers
382 views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
9
votes
8answers
263 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
5
votes
2answers
372 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
2
votes
3answers
98 views

Help with a proof that sequence of rational numbers $ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converges to an irrational, $\sqrt2$

I know that there are sequences of rational numbers with irrational limits. One in particular I've seen is $$ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ with $a_0 =1$, This is clearly rational ...
13
votes
3answers
202 views

For $x_{n+1}=x_n^2-2$, show $\lim_{n\to\infty}\frac{x_n}{x_0x_1\cdots x_{n-1}}=2$

Suppose $x_0:=2\sqrt{2}$ and $x_{n+1}=x_n^2-2$ for $n\ge1$. We have to show $$\lim_{n\to\infty}\frac{x_n}{x_0x_1\cdots x_{n-1}}=2$$ Establishing convergence is pretty direct but I'm having trouble ...
1
vote
0answers
43 views

Monotonicity of a recursively defined sequence $s_{n+1} = \sqrt{4s_n -1}$

I am trying to answer this question: Let $s_1 = k$ and $s_{n+1} = \sqrt{4s_n -1}$. For what values of $k$ will the sequence $s_n$ be monotone increasing? I know the definition of monotone increasing ...
3
votes
3answers
180 views

Proof Using the Monotone Convergence Theorem for the sequence $a_{n+1} = \sqrt{4 + a_n}$

Consider the recursively defined sequence $a_0 = 1$ $a_{n+1} = \sqrt{4 + a_n}$ How to prove that the sequence converges using the Monotone Convergence Theorem, and find the limit?
2
votes
2answers
68 views

Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...
17
votes
1answer
221 views

Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
0
votes
1answer
44 views

$a_1=3$ and $a_{n+1}=\dfrac{2a_n}{3}+\dfrac{4}{3a_n^2}$. Show that $4^{1/3} \le a_n$for all $1\le n$.

$a_1=3$ and $a_{n+1}=\dfrac{2a_n}{3}+\dfrac{4}{3a_n^2}$ By considering the function $f(x)=\dfrac{2x}{3}+\dfrac{4}{3x^2}$, show that $4^{1/3} \le a_n$ for all $1\le n$.
0
votes
1answer
114 views

Limits of a recursively defined sequence [closed]

Let $x_1=a$ and define a sequence $\left(x_n\right)$ recursively by: $x_{n+1} = \dfrac{x_n}{1 + \frac{x_n}{2}}$ For what values of $a$ is it true that $x_n$ approaches $0$?
3
votes
4answers
204 views

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent.

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent. What is the limit? The back of my textbook says that ...
2
votes
2answers
364 views

Nested roots sequence, how to prove it's monotone and bounded?

Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$ Here's what I did: Plugging in some values makes it seem as if the sequence is increasing. I ...
8
votes
2answers
1k views

Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct?

Problem The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation: $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Evaluate the limit $\lim_{n\to\infty} a_n$. Solution Show that the sequence ...
2
votes
1answer
146 views

recurrence relations for proportional division

I am looking for a solution to the following recurrence relation: $$ D(x,1) = x $$ $$ D(x,n) = \min_{k=1,\ldots,n-1}{D\left({{x(k-a)} \over n},k\right)} \ \ \ \ [n>1] $$ Where $a$ is a ...
3
votes
6answers
83 views

How to prove a limit with a recurrence?

$s_1 = 1$ and $s_{n+1} = \dfrac{s_n + 1}{3}$ for $n \in \Bbb N$. How do you find $\displaystyle \lim_{x\to \infty} s_n$? Then how do you prove that the value is the limit using the definition of the ...
15
votes
5answers
633 views

limit of : $a_{n+2} =\frac{1}{a_n} + \frac{1}{a_{n+1}}$

$a_n$ is a real sequence, $a_1,a_2$ are positive and for all $n>2$ : $$ a_{n+2} =\frac{1}{a_{n+1}} + \frac{1}{a_{n}}.$$ Prove that: $\displaystyle \lim_{n\to \infty} a_n$ exists, then find it. ...
1
vote
1answer
75 views

Convergence of $x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$

$x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$ with $x_0=a\ge0$ I calculated some values for $x_n$ with $a=0,1,2$ and recognized its monotonically decreasing (It converges to $0$). Now the standard approach ...
6
votes
5answers
148 views

$x_{n+1} = \sqrt{3x_n}$ converges for $x_1 = 1,x_1 = 27$ Proof

Prove $x_{n+1} = \sqrt{3x_n}$ converges for $x_1 = 1,x_1 = 27$. (Separate problems for $x_1 = 1$ and $x_1 = 27$.) EDIT: Took out bad algebra.
20
votes
4answers
508 views

Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$

Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical $$ \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}} ...
1
vote
0answers
73 views

Let $f(x+1) = P(f(x),x)$ where P is a polynomial. Express $f(x)$ as an integral.

Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial. Express $f(x)$ as an integral from $0$ to $\infty$. As an example we ...
3
votes
1answer
884 views

Dyadic Expansion-Proof?

Working through a measure theory textbook, and would like to understand dyadic expansions before I can understand its connections with the law of large numbers. I want to see this proved in detail, ...
12
votes
2answers
506 views

Zeroes of the third derivative of an iterated sine.

I've been playing with the functions $$f_n:[0,\pi/2]\to[0,1]\\\begin{cases} f_1&=&\sin\\f_{n+1}&=&\sin\circ f_n\end{cases}.$$ A simple argument proves that $f_n(x)\to 0$ for $x\in ...
2
votes
4answers
140 views

Proving an exponential bound for a recursively defined function

I am working on a function that is defined by $$a_1=1, a_2=2, a_3=3, a_n=a_{n-2}+a_{n-3}$$ Here are the first few values: $$\{1,1,2,3,3,5,6,8,11,\ldots\}$$ I am trying to find a good approximation ...
2
votes
2answers
467 views

Limit of a recursive sequence

Let $\lambda$$\in$$(0,1)$. For any real $a_0$, $a_1$, define the sequence recursively by $$a_n = (1-\lambda)a_{n-1} + \lambda a_{n-2}$$ Let $\alpha$ = $\lim\limits_{n\rightarrow\infty}a_n$ Express ...