6
votes
1answer
125 views

Limit related to the recursion $a_{n+1}=(1-n^{-1/4}a_n)a_n$

Let $0<a_{1}<1$, with $$a_{n+1}=a_{n}(1-n^{-\frac{1}{4}}a_{n})$$ Does there a real numbers $A$ which make the limit $$ ...
0
votes
0answers
65 views

Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
6
votes
2answers
89 views

Is the sequence $x_{n+1} = \frac{x_n + \alpha}{x_n + 1}$ convergent?

Fix $\alpha > 1$, and consider the sequence $(x_n)_n \geq 0$ defined by $x_0 > \sqrt \alpha$ and $$x_{n+1} = \frac{x_n + \alpha}{x_n + 1}, n = 0, 1, 2, \dots$$ Does this sequence converge, ...
1
vote
1answer
21 views

Cardinality of a set with a recurrence relation.

Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$ What is $\left|A\right|$? Well, I tried to treat $f$ as a recurrence ...
1
vote
3answers
37 views

Finding an exact solution to a difference equation

How would I solve an equation of the form: $u(n+1)=1/2u(n)+(1/3)^n$ when $u(0)=1$? I got an answer of the form $u(n)= c + \sum(1/3)^j*2^{j-1}$ but believe this is incorrect?
0
votes
3answers
57 views

Finding a reduction formula for this integral

Let $$I(n)=\int_0^1 (x-x^2)^n dx .$$ Mainly, what I'm trying to get is a recurrent form of this integral that probably involves $I(n-1)$. My attempt ...
1
vote
2answers
39 views

Recurrence with multiplication

Let $\{a_{n}\}$ be a sequence of nonnegative numbers such that $a_{n} = 2^{n}a_{n - 1}^{3/2}$. If $a_{1}$ is sufficiently small, why must $a_{n} \rightarrow 0$ as $n \rightarrow \infty$?
3
votes
1answer
76 views

UNBEATABLE recurrence relation

Hi I don't know where to start to solve this reccurence relation: $g(1)=2$ $ g(2n)=3g(n)+1$ $ g(2n+1)=3g(n)-2$ of coures I can make it: $ g(1)=2$ $ g(n)=g(2n)/3-1/3$ $ ...
0
votes
0answers
36 views

Total Time for a ball that bounces from a height of 8 feet and rebounds to a height 5/8 [duplicate]

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
0
votes
2answers
92 views

Total Time a ball bounces for from a height of 8 feet and rebounds to a height 5/8

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
6
votes
2answers
586 views

Recursive square root problem [duplicate]

Give a precise meaning to evaluate the following: $$\large{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dotsb}}}}}$$ Since I think it has a recursive structure (does it?), I reduce the equation to $$ ...
0
votes
1answer
124 views

Backwards Recurrence [duplicate]

For $$I_n = \int_0^1 \frac{x^n}{x + \alpha}\,{\rm d}x$$ $\alpha$ constant parameter We know $I_0$ = log[ $\frac{1 + \alpha}{\alpha}$] and we can derive a recurrence formula for $I_n$: $I_n = \frac 1n ...
1
vote
3answers
106 views

$W_n=\int_0^{\pi/2}\sin^n(x)\,dx$ Find a relation between $W_{n+2}$ and $W_n$

Set $$W_n=\int_0^{\pi/2}\sin^n(x)\,dx.$$ Compute $W_0$ and $W_1$. Find a relation between $W_n$ and $W_{n+2}$ and deduce a formula for $W_n$. What I have so far is: $$W_{2k}=\frac{1}{2^k}\left( ...
1
vote
1answer
53 views

Convergence of a sequence with repeated sines [duplicate]

Let $x\in(0,\pi/2)$ and $\{a_n\}_{n\in\mathbb N}$ defined recursively as follows: $$ a_0=x, \quad \text{and} \quad a_{n+1}=\sin(a_n). $$ Show that $$ \lim_{n\to\infty}{n\,a_n^2}=3. $$ Note. There is ...
1
vote
1answer
28 views

Proving the monotonicity of a recurrence.

Define the following recurrence for $n = 1, 2, \cdots$ $T(n) = ( 1 - \operatorname{H}(\frac{1 - P^{\frac{1}{n}}}{2}))^n$ where $0 < P < 1$ is a constant, function $\operatorname{H}(\cdot)$ is ...
0
votes
0answers
36 views

limit of a recursive sequence, Am I allowed to divide by $b_n^2$?

$$b_1 > 0$$ $${b_{n + 1}} = {{{b_n}^2 + 1} \over {{b_n}}} = {{{{{b_n}^2} \over {{b_n}^2}} + {1 \over {{b_n}^2}}} \over {{{{b_n}} \over {{b_n}^2}}}}\mathop = {{1 + {1 \over {{b_n}^2}}} \over {{1 ...
2
votes
4answers
121 views

Recurrence sequence limit

I would like to find the limit of $$ \begin{cases} a_1=\dfrac3{4} ,\, & \\ a_{n+1}=a_{n}\dfrac{n^2+2n}{n^2+2n+1}, & n \ge 1 \end{cases} $$ I tried to use this - $\lim ...
2
votes
1answer
63 views

Recurrence relation of an integral [closed]

$$ y_n = \int_0^1 \frac{x^n}{x+5} dx,\ \ n=0,1,2,...,\infty $$ What is the recurrence relation between $y_n, y_{n-1}$ ? Could you provide me a methodology on how to proceed with this?
0
votes
1answer
27 views

Linear homogeneous recursive sequence of constant sign

Let recursive sequence be defined by the formula $$ s_{j+1}=as_j-s_{j-1}, $$ where $a>1$ is some integer number. Is it true that $s_0<0$, $s_1<0$ implies $s_j<0$ for $j \geq 0$? Edit: ...
0
votes
1answer
42 views

recurrence relation question.

I have this expression: $I_{n} = \int_0^1 \frac{x^{n-1}}{2-x} dx$ for $n=1,2,3,\ldots$ I have been asked to show that by writing $x^n = x^{n-1} (2-(2-x))$ that the recurrence relation $I_{n+1} = 2I_n ...
0
votes
1answer
68 views

Solving a recurrence with division

I'm having this recurrence that is giving me a lot of trouble. $$F(2^0) = 1$$ $$F(2^k) = \frac 1 2 F(2^{k-1}) + 2^k$$ I will edit my post since this does not seems to work... The initial ...
10
votes
1answer
111 views

Find asymptotic of recurrence sequence

Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$. The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$. ...
1
vote
4answers
1k views

Find the limit of a recursive square root sequence.

Find the limit of the sequence $$\left\{\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots\right\}$$ Another way to write this sequence is $$\left\{2^{\frac{1}{2}},\hspace{5 pt} ...
6
votes
1answer
103 views

How to prove that this recursively defined sequence converges to $e$?

Let $a_1=0,a_2=1,$ and $a_{n+2}=\dfrac{(n+2) a_{n+1}-a_n}{n+1}$. Prove that $\lim_{n\to \infty}a_n=e$. I know that $\lim_{n\to\infty}\left(1+\frac1{2!}+\frac1{3!}+...+\frac1{n!}\right)=e$ and $a_n = ...
0
votes
1answer
66 views

How to compute the formula of $S_n$

$S_1$=a, $S_2$=b, $S_n$=|$S_{n-1}$-$S_{n-2}$|(n $\ge$3). Can I compute the formula of $S_n$? Thanks in advance.
27
votes
2answers
961 views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...
1
vote
2answers
114 views

Recurrence relation: $x_{n+1} = \frac 12 x_n + \frac 1 {x_n},$ [duplicate]

$$x_{n+1} = \frac 12 x_n + \frac 1 {x_n}, x_0 \neq 0$$ $$ a = \frac a2 + \frac 1a \Rightarrow a = \frac {a^2 + 2} {2a} \Rightarrow 2a^2 = a^2 + 2 \Rightarrow a^2 = 2 \Rightarrow a = \pm \sqrt 2$$ If ...
0
votes
3answers
111 views

Recurrence relation with periodic function

$$x_{n+1} = x_n + \sin x_n$$ $$x_{n+1} = \sin \left(\frac {\pi} {2} x_n\right)$$ How to solve these? Or, at least, what can be said about thier behavior and limits?
1
vote
5answers
86 views

solution of recurrence relation $x_{n+1} = \frac {1}{x_n + 1}$

$$x_{n+1} = \frac {1}{x_n + 1}; x_1 > 0$$ How to transform it into the form $x_n = $? I need the solution in order to check if it converges at any $x_1 > 0$.
2
votes
1answer
61 views

How to solve $x_{n+1} = \frac{x^2_n + 1}{x_n}$ if $x_0>1$?

How to solve the following recurrence relation, assuming that $x_0 > 1$: $$x_{n+1} = \frac{x^2_n + 1}{x_n}$$ Am I allowed to divide the fraction, that is $x_{n+1} = x_n + \frac{1}{x_n}$?
0
votes
2answers
72 views

recurrence relation: $x_{n+1} = x^2_n - 2x_n + 2$

$$x_0 = \frac32; x_{n+1} = x^2_n - 2x_n + 2$$ $$\Rightarrow x = x^2 - 2x +2 \Rightarrow x^2 - 3x +2 = 0 \Rightarrow x = {1;2} $$ How to determine which one is the limit, i.e. $\lim_{n\rightarrow ...
2
votes
2answers
473 views

A recursive formula for $a_n$ = $\int_0^{\pi/2} \sin^{2n}(x)dx$, namely $a_n = \frac{2n-1}{2n} a_{n-1}$

Where does the $\frac{2n-1}{2n}$ come from? I've tried using integration by parts and got $\int \sin^{2n}(x)dx = \frac {\cos^3 x}{3} +\cos x +C$, which doesn't have any connection with ...
20
votes
2answers
402 views

How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?

I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$, where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$. Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...
32
votes
1answer
649 views

Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$

If $S = x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Find S. Note:This is not a GP series.The powers are in GP. My Attempts so far: 1)If $S(x)=x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Then ...
3
votes
1answer
243 views

Using a definite integral, to create a specific recurrence relation.

Hello i have the integral: $$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$ I need to show that the integral can be represented by the recurrence relation below; $$y_n= ...
12
votes
2answers
419 views

Finding the limits

Suppose $a_1=1, a_{k+1}=\sqrt{a_1+a_2+\cdots +a_k}, k \in \mathbb{N}$. Find the limits $$i)\space \lim_{n\to\infty}\displaystyle \frac{\sum_{k=1}^{n} a_{k}}{n\sqrt{n}}$$ $$ii)\space ...
2
votes
3answers
80 views

Finding a general solution of $A_n$

Find the general solution to $ A_{n+1} + 4A_n = n $ I am unsure how to even start the question :S
2
votes
2answers
75 views

What's $T\left(n\right)$?

If $T\left( n \right) = 8T\left( n-1 \right) - 15T\left( n-2 \right); T\left(1\right) = 1; T\left( 2 \right) = 4$, What's $T\left(n\right)$ ? I use this method: Let $c(T(n) - aT(n-1)) = T(n-1) - ...
2
votes
4answers
1k views

Finding the limit of a recursive sequence $x_{n+1}=\frac{1}{2}(x_{n}+\frac{M}{x_{n}})$

Define the sequence $\{x_n\}_{n\in\mathbb{N}}$ by $$x_0=\frac{M}{2}, \qquad x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{M}{x_{n}}\right)$$ where $M\in\mathbb{R}$, $M\geq 0$. Find ...
-3
votes
1answer
155 views

a recursive and dificult sequence [duplicate]

Possible Duplicate: How to prove that this sequence converges? Let the sequence defined recursively by the equation: $$ a_n = a_{a_{n - 1} } + a_{n - a_{n - 1} } $$ How can I prove ...
8
votes
2answers
548 views

Reduction formula for $I_{n}=\int {\cos{nx} \over \cos{x}}\rm{d}x$

What would be a simple method to compute a reduction formula for the following? $\displaystyle I_{n}=\int {\cos{nx} \over \cos{x}} \rm{d}x~$ where $n$ is a positive integer I understand that it ...