For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0
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1answer
48 views

Deciphering the main theorem of the paper ''On Oblath's Problem''

I am trying to read the paper On Oblath's Problem, and I'm have difficulty understanding the main theorem. I can read the theorem but I don't understand it. May someone help me to make this theorem as ...
7
votes
3answers
337 views

How can I rewrite recursive function as single formula?

There is following recursive function $$ \begin{equation} a_n= \begin{cases} -1, & \text{if}\ n = 0 \\ 1, & \text{if}\ n = 1\\ 10a_{n-1}-21a_{n-2}, & \text{if}\ ...
1
vote
1answer
42 views

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of ...
5
votes
1answer
69 views

Evaluating sums of the form $\sum_{i_d=1}^{\infty}\ldots\sum_{i_2=1}^{\infty}\sum_{i_1=1}^{\infty}x^{i_1\cdot i_2\cdots i_d}$

I am wondering if there is a way to evaluate or get a more useful expression for a sum of the following form: $$\sum_{i_d=1}^{\infty}\ldots\sum_{i_2=1}^{\infty}\sum_{i_1=1}^{\infty}x^{i_1\cdot i_2\...
0
votes
3answers
84 views

The sequence defined by $x_{n}=\frac{x_{n-1}+x_{n-2}}{2}.$

Let $x_{1}=0,x_{2}=1$ and for $n\geq3,$ define $x_{n}=\frac{x_{n-1}+x_{n-2}}{2}.$ Which of the following is/are true? $1.\{x_{n}\}$ is a monotone sequence. $2. \lim_{n\to\infty} x_{n}=\frac{1}{2}.$ ...
2
votes
4answers
29 views

Convergence and Limit of a Recursive Sequence from the multiples of $a_n$

I'm having trouble with this recursive sequence problem. I'm supposed to find the limit, assuming that is it convergent, but I can't seem to get the answer. $a_1 = 1, a_{n+1}= \frac {2a_n}{7+a_n} $
4
votes
3answers
135 views

Closed form for $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$

Is there any closed form for the infinite product $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$? I think it is convergent for any $x\in\mathbb{R}$. I think there might be one because there is a closed form ...
1
vote
1answer
79 views

Graphing $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$: help please!

I am investigating the sequence that tends to the limit $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$, and although I am making headway on related theory, I would like to graph the ...
0
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0answers
30 views

maths question on combination [closed]

16 teams took part in a win-lose competition playing both home and away matches against each other for 30 days. if 8 matches are played concurrently in a day with two teams playing against each ...
0
votes
0answers
33 views

help on proving converging of sequence, please

I have $f(x)$. I know this $$|f'(x)|≤ C < 1 $$ $$0≤C$$ $$x ∈R$$ and I have this sequence $$a_1=x\\ a_2=f(x)\\ a_3=f(f(x)),\\\vdots\\ a_n=f(\cdots(f(x))\cdots)$$ I need to prove this sequence ...
3
votes
1answer
185 views

help verifying equation $\int_0^ x \frac{1}{1+t^n} dt$

As a follow up to a previous posting addressing the integral of $1/ (t^n+1)$ for $n\in \Bbb{N}$ I found the following $$\int_0^ x \frac{1}{1+t^n}\, dt=\sum_{i=0}^{\infty}\frac{(i!)(n^i)x^{in+1}} {(x^...
2
votes
3answers
44 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
-1
votes
0answers
53 views

help in proving converge of this sereis, please??? [closed]

I have $f(x)$. I know this $$|f'(x)|≤ C < 1 , 0≤C , x ∈R and I have this sequence $$a_1=x\\ a_2=f(x)\\ a_3=f(f(x)),\\\vdots\\ a_n=f(\cdots(f(x))\cdots)$$ I need to prove this sequence ...
0
votes
2answers
66 views

Transform double sum $\sum_{i=0}^\infty \sum_{j=0}^i$ to $\sum_{i=0}^\infty \sum_{j=0}^\infty$?

Consider a double sum (assuming it converges) $$\sum_{i=0}^\infty \sum_{j=0}^i f(i,j)$$ Is there a convenient way to rewrite this sum so that both summations go from zero to infinity $\sum_{i=0}^\...
0
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0answers
27 views

The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows \begin{equation}\label{cp26} F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...
5
votes
1answer
116 views

How do I find a closed form of ${\pi^{2n}\over \zeta(2n)}\int_{-1}^{1}{x^{2n-2}\over \pi^2+(2\tanh^{-1}{x})^2}dx$?

How do I evaluate the closed form for $g(n)$? Where n is an integer, $n\ge 1$ $${\pi^{2n}\over \zeta(2n)}\int_{-1}^{1}{x^{2n-2}\over \pi^2+(2\tanh^{-1}{x})^2}dx=g(n)$$ Make a subsititution $u=\...
4
votes
2answers
91 views

Is boundedness required in equivalence between $\frac1n\sum_{k=1}^na_k\to0$ and $\frac1n\sum_{k=1}^na_k^2\to0$?

Suppose $a_n$ is a sequence of non-negative real numbers. If $a_n$ are un-bounded, then I want to know if $\dfrac{1}{n}\sum_{k=1}^na_k\to0$ as $n\to\infty$ is equivalent to $\dfrac{1}{n}\sum_{k=1}^...
6
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3answers
130 views
+50

A System of Infinite Linear Equations

Suppose that $\{a_{i}\}_{i=-\infty}^{\infty}$ with $\sum_{i=-\infty}^\infty a_{i} \lt \infty$ is known and that $\{b_i\}_{i=-\infty}^{\infty}$ is such that $$\sum_{i=-\infty}^\infty a_{i}b_{-i} =1,$$ ...
1
vote
1answer
43 views

Interval of convergence for series with complex numbers

I'm trying to find interval of convergence of this series: $$\sum_{n=1}^{\infty} \frac{7^n(z+2i)^n}{4^n+3^ni}$$ and I should draw a plot which represents the answer, this is what I've got so far: ...
0
votes
3answers
81 views

Convergence problem $\sum \left(1-n\sin\left(\frac{1}{n}\right)\right)$ [closed]

I have to check convergence of: $$\sum_{n=1}^\infty\left(1-n\sin\left(\frac{1}{n}\right)\right).$$ I have no idea but I only check that $\lim \ n\left(1-n\sin\left(\frac{1}{n}\right)\right)=0$.
2
votes
0answers
34 views

Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
2
votes
3answers
77 views

Convergence of $\sum a^{1/x_n}$ for $a$ in $(0,1)$ and $\sum x_n$ a positive convergent series

Let $\sum x_n$ be a convergent series of positive real numbers and $0<a<1 $, then is the series $\sum a^{1/{x_n}}$ convergent ? I have only figured out that $\lim a^{1/{x_n}}=0$.
0
votes
1answer
62 views

Power series / Taylor series approximation

I need to find $k\in N$ such that $$ \dfrac{k}{10\ ^ 5} \le \arctan(0.1) \lt \dfrac{k +1}{10\ ^ 5} $$ I tried using Lagrange Remainder formula to find that k but with no luck. If I am using ...
0
votes
1answer
46 views

Fibonacci Sequence: Prove $f_1+f_3+\dots+f_{2n-1}=f_{2n}$ by Induction.

I believe the majority of my proof is correct I'm just not certain about the base case if any one can explain how to do that base case or fix any error I made I would greatly appreciate it. Recall ...
5
votes
0answers
84 views

Integral with an infinite sum

Let: $$\mathcal{S}(x)=\sum_{k=1}^{\infty}-\frac{\cos(\frac{\pi}{2}+k x)}{k^x}$$ I need help evaluating $$\int_{0}^{1}\mathcal{S}(x) dx$$ Obviously the cosine term in the numerator simplifies to $\...
1
vote
5answers
136 views

Sum of $1^2+3^2+\cdots+(2n+1)^2$ Have trouble with proof.

I've been working through a question on Courant's What is Mathematics? This is the question: Prove $1^2+3^2+\cdots+(2n+1)^2=\frac{n(n+1)(2n+1)(2n+3)}{3}$. I called this $S_{(2n+1)^2}$. What I've ...
2
votes
2answers
24 views

Is the Cesàro Sumation of Series divergent to infinity divergent?

More specifically the question is: If I have a series $(u_n)_{n\in\mathbb{N}} \subset \mathbb{R}$ that diverges to infinity. Then it's cesaro sumation series $(s_n)_{n\in\mathbb{N}}=(\frac{1}{n+1}\...
2
votes
0answers
35 views

Why are sequences and functions notated differently?

Why do we usually write, e.g., $s_n$ for sequences, while functions are usually written as $f(x)$? Conceptually, aren't sequences just functions with a subset of the naturals, not of the reals, as ...
3
votes
5answers
129 views

Does the series $\sum_{n=1}^\infty\frac{n^{\sqrt{n}}}{n!}$ converge?

Immediately I recognize that there's a factorial and I use the ratio test to try and solve it: $$\lim_{n \rightarrow \infty}\left|\frac{{(n+1)}^{\sqrt{n+1}}}{(n+1)!}\cdot\frac{n!}{n^{\sqrt{n}}}\right|...
0
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0answers
33 views

Continous functions and zeros

How to prove following theorem? If sequence $\{f_n\}$ of continous real functions with domain $D \subset \mathbb{R}$ is compact convergent to $f$ and sequence $\{x_n\}$ with $D$ satisfies $f_n(x_n) = ...
1
vote
1answer
38 views

I am trying to prove the distribution function for the 'birthday problem' can anyone help?

Let $Y_1, Y_2, . . .$ be i.i.d. and uniformly distributed on the set ${1, 2, . . . , n}$. Define $X^{(n)} = \min \{k : Y_k = Y_j \,\,for \,\,some \,\,j < k\} $, the first time that we see a ...
0
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1answer
41 views

Variant of the geometric series

Can someone explain me how one computes $$\sum_{k=1}^n kq^k = \dfrac{nq^{n+2}-(n+1)q^{n+1}+q}{(1-q)^2}$$ and what exactly the derivative has to do with it?
0
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0answers
55 views

Find the limit of sequence.

Assume $ f(x) \in C^2[a,b] $ and $f(a)f(b)<0,f'(x)>0,f''(x)>0,\forall x \in[a,b]. $Prove the sequence $$ x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n) } \quad x_1 \in[a,b]\text{ and } x_1 \text{is to ...
5
votes
5answers
112 views

Bounding a series: $\frac{\pi}{2} < \sum_{n=0}^\infty \frac{1}{n^2 + 1} < \frac{3\pi}{2} $

I have the following statement - $$\frac{\pi}{2} < \sum_{n=0}^\infty \dfrac{1}{n^2 + 1} < \frac{3\pi}{2} $$ So I tried to prove this statement using the integral test and successfully proved ...
1
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0answers
33 views

Binomial Series Expansion and then find an approximate value [closed]

This is from an A-Level Maths paper. Show that $\frac{x}{(1-x)^3} = x + 3x^2 + 6x^3 + O(x^4)$ (Only first three terms of infinite series expansion are asked for) Use the result to find an approximate ...
2
votes
1answer
40 views

Where is the mistake of a possible application of Frullani's theorem in this case?

My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ...
0
votes
2answers
56 views

What is the relation between $x,y$ if $\tan(20^\circ),x,\tan(50^\circ)$ and $\tan(20^\circ),y,\tan(70^\circ)$ are in AP?

If $\tan(20°),x,\tan(50°)$ are in AP and $\tan(20°),y,\tan(70°)$ are in AP then relation between x and y is?. $$\text{Attempt}$$. As they are in AP So $2x=\tan(20°)+\tan(50°),2y=\tan(20°)+\tan(70°)$ ...
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votes
1answer
118 views

Telescoping function Revealed. [on hold]

Part B: I found Summation $$\sum_{n=0}^\infty\frac{x^n \ (-1)^n}{(y+1)^{n+1}} = \frac{1}{(x+y+1)}$$ However $x$ is related to $y$. $y \ge |x|$. You may derive the result and present it by ...
1
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0answers
58 views

Limit of $\sum_{k=0}^{n}\frac{1}{2k+n}$ and similar

Examine wether following sequences have limits and if yes - find them. a)$\sum_{k=0}^{n}\frac{1}{2k+n}$ b)$\sum_{k=0}^{n}\frac{(-1)^n}{2k+n}$ c)$\sum_{k=0}^{n}\frac{(-1)^k}{2k+n}(\frac{1}{3})^k$ a)...
4
votes
1answer
174 views

New series formula for $\arctan(x)$?

I discovered this equation, but have no idea if it has been previously discovered. Please help determine if it has been previously developed. Or please prove that the equation is not correct. $$\...
0
votes
2answers
43 views

How to show the sequence is monotone

"$u_n = \frac{2}{1+e^{-n}}$. Show that $u_n$ is monotone." My approach would be to consider |$u_{n+1} - u_n$| = |$\frac{2}{1+e^{-n-1}} - \frac{2}{1+e^{-n}}$|. However I'm not sure the best way to ...
2
votes
3answers
53 views

prove that $\sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)}$ is convergent and find the limit when $n \to \infty$

does the following sumatory converges? if yes find the limt when $n \to \infty$ $$\sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)}$$ ideas? i have tried by the comparison test.
2
votes
1answer
27 views

Greatest number in fibonacci sequence with property: sum of digits=index in fibonacci sequence

I came across very interesting sequence based on fibonacci sequence. From fibonacci numbers we choose only elements with digit sum=index in fibonacci sequence. It is very interesting that we most ...
19
votes
5answers
3k views

A very curious rational fraction that converges. What is the value?

Is there any closed form for the following limit? Define the sequence $$ \begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$ with initial values $a_0 = b_0 = 1$. ...
2
votes
1answer
59 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
4
votes
4answers
200 views

How to prove that the series $\sum\limits_{n=1}^\infty {\sin^2n} $ diverges

I want to use a divergence test to prove that $\lim_{n\to \infty} \sin^2n$ does not converge. So $\sum_{i=1}^\infty \sin^2 n $ diverge. But because $\pi$ is an irrational number. So I cannot use ...
0
votes
0answers
34 views

Find the next numbers in the sequence.. [closed]

What is the next number in the sequence: 11, 67, 348, 1071, .... I tried factorizing and taking differences but I can't find common relationship in the sequence.
0
votes
1answer
39 views

$\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
0
votes
3answers
30 views

bounds on a sequence

It may look that this question is trivial, but: Let $(a_n)_{n=1}^\infty$ a sequence s.t. $\forall n\in \mathbb{N} \ \ a_n<\frac {1}{n}$. Prove/Disprove : there is $c > 1$ s.t. $\forall n\in \...