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

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-1
votes
0answers
24 views

How do I identify pattern of different bases? For example, base $3$?

Recently,I asked a question on number patterns.Turns out it wasn't the usual number pattern question.It was the sum of the digits at base 3. How do I identify such number patterns or even know that ...
0
votes
1answer
15 views

Necessary and/or sufficient conditions for summability of a sequence

It is clearly true that any $(a_n)_{n=1}^\infty$ that has $$a_n=O(n^{-1-\varepsilon}),$$ for some fixed $\varepsilon>0$, is absolutely summable: $$\sum\limits_{n=1}^\infty |a_n|<\infty.$$ My ...
4
votes
1answer
82 views

Daunting series of integrals

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
2
votes
1answer
59 views

What is the $2012th$ number in this pattern?

This is question 30 from Australian Maths 2012 $(0,1,2,1,2,3,2,3,4,1,2,3,2,3,4,3,4,5,2,3,4,...)$ What is the $2012th $ number in this list? What I did: I broke up the first few numbers into ...
1
vote
3answers
58 views

Does this $\sum_{n=1}^\infty(-1)^n\tan(\frac{\pi}{n+2})\sin(\frac{n\pi}{3} )$ converge and why? [on hold]

Does the series $\sum_{n=1}^\infty(-1)^n\tan(\frac{\pi}{n+2})\sin(\frac{n\pi}{3} )$ converge and why?
0
votes
1answer
16 views

Find all radiuses of convergence for this series - is my approach correct?

I'm supposed to find all radiuses of convergence for this power series: $\sum_{k=0}^{\infty} \frac{k^{2}}{3^{k}}x^{k}$ I've worked with ratio test: $\frac{{}\frac{(k+1)^{2}}{3^{k+1}}}{\frac{k^...
1
vote
1answer
31 views

If a series can be rearranged to sum to N different values, can it be rearranged to sum to any value?

This question made me think of a related question: Suppose we have a sequence $a_n$ and a set of permutations $S$ with $|S| = N$ (for some $N > 1$). Suppose that, for any two (distinct) ...
1
vote
2answers
31 views

What's the series and what's the radius of convergence of this (power) series?

Find the convergence radiuses of this power series: $1 + n + n^{4} + n^{9} + n^{16} + n^{25} + n^{36} + ...$ First of all, I'm surprised it says $radiuses$ instead of $radius$. I know you find ...
2
votes
1answer
47 views

How do I show that $E-\gamma=\lim_{j\to \infty}\sum_{n=1}^{j}n\left({1\over 2^n-1}-{1\over 2^n}+\cdots-{1\over 2^{n+1}-2}\right)$

Given the Erdos-Borwein's constant $E=\sum_{n\geq 1}\frac{1}{2^n-1}$ and the Euler-Mascheroni constant $\gamma=0.5772156...=\sum_{n\geq 1}\left[\frac{1}{n}-\log\left(1+\frac{1}{n}\right)\right]$ how ...
3
votes
2answers
258 views

Find a formula for a sequence

I'm trying to find a formula for the following sequence: $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$ I thought of solving it recursively and I got this formula: $a_{n}=\sqrt{3*a_{n-...
1
vote
0answers
31 views

How does one visualize Abel's test?

I remember that at a lecture we had a visual representation of why Abel's test should work but I don't recall how exactly it was. It was somehow representing $a_{n}$ on one axis and $b_{n}$ on another ...
2
votes
2answers
134 views

A series that can be rearranged to converge to any number converges conditionally

Riemann's theorem states that if a series is conditionally convergent, then for any number $L$ (could be infinite), the series can be rearranged in such manner that it would converge to $L$. I was ...
1
vote
1answer
28 views

Series involving modified Bessel functions and sine/cosine functions

I would like to understand if the following two infinite series can be further expressed in terms of known functions: $$\sum_{n\geq 0} (-1)^nI_{2n+1}(A)\frac{\cos((2n+1)B)}{2n+1}$$ and $$\sum_{n\...
1
vote
2answers
33 views

How to determine the convergence of the following series?

Good evening to everyone! I have the following series $$ \sum _{n=1}^{\infty }\left(-1\right)^{n }\left|\alpha -1\right|^n\frac{n!}{\left(n+1\right)!-n!+1} $$. I don't know from where to start to ...
-3
votes
1answer
56 views

What did I do Wrong Here?

I believe I have made a mistake in the following work I did: I have no idea if i have made a mistake or, if somehow, i did everything right. So explain what my mistake is.
1
vote
1answer
49 views

Prove convergence of a sequence.

Let $a_n$ be a series of non-negative real numbers. Suppose $\sum a_n $ diverges. Prove : If $\lim(na_n)$ exists (in $\mathbb R$ or $\infty$), then $\sum \dfrac{a_n}{1+na_n}$ diverges. Thanks ...
3
votes
2answers
33 views

Hints on showing Cauchy sequence converges

Let $T>0$ and $L\geq0$. Let $C[0,T]$ be the space of all continuous real valued functions on $[0,T]$ with the metric $\rho$ defined by $$\rho(x,y)=\sup_{0\leq t\leq T}e^{-Lt}\left|x(t)-y(t)\right|$...
4
votes
1answer
26 views

Find the radius of convergence of this power series: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$

Given: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$ I started by forming it: $\binom{2k}{k} = \frac{(2k)!}{k!*(2k-k)!} = \frac{(2k)!}{k!*k!}$ Now the problem is, I cannot write $2! * k!$ instead of $(...
2
votes
1answer
15 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
0
votes
1answer
26 views

Three odd Three even

I want a formula that generates evens when $x\equiv 1\pmod 6$,$x\equiv 2\pmod 6$and$x\equiv 3\pmod 6$and also generates odds when $x\equiv 4\pmod 6$,$x\equiv 5\pmod 6$and$x\equiv 5\pmod 6$. This is ...
0
votes
0answers
36 views

On oblath's theorem [on hold]

it is just my first encounter about this topic ,it is the topic that my prof gave to me in my undergrad studies.I found it interesting but there are still parts(like theorem) in this topic which make ...
2
votes
2answers
59 views

Closed form for an integral with log and power

Let $n \in \mathbb{N}$. We know that: $$\int_0^1 x^n \log(1-x) \, {\rm d}x = - \frac{\mathcal{H}_{n+1}}{n+1}$$ Now, let $m , n \in \mathbb{N}$. What can we say about the integral $$\int_0^1 x^n \...
1
vote
0answers
27 views

On Oblath's Problem [on hold]

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 ...
-1
votes
1answer
26 views

Problem involving sequence of random variables on probability space [on hold]

How do I construct (and prove that) an example of a sequence of random variables $\{X_n\}_{n\, \ge\, 1}$, on an appropriate probability space, for which $X_n$ converges to $0$ in $L^r$ for all $r > ...
-4
votes
1answer
77 views

new equation for $\int_0^ t e^{-x2} dx$? [on hold]

fact! $$\int_0^ x e^{-x^2} dx$$ $$=e^{-x^2}\sum_{n=0}\frac{(2^n)x^{2n+1}}{{(2n+1)!!}}$$
0
votes
1answer
46 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
324 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
68 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
82 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
127 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
77 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
votes
0answers
30 views

maths question on combination [on hold]

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
30 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 ...
1
vote
1answer
135 views

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

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
43 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??? [on hold]

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
59 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
votes
0answers
26 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
110 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
89 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}^...
5
votes
1answer
63 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
0answers
23 views

Convergence of a big sum indexed over $\mathbb{Z}^3$

For a fixed vector $r_j$ consider the function on $\mathbb{R}^3$ defined by the series $$f(r) = \sum_{\substack{n,m,k \in \mathbb{Z} \\ (n,m,k) \neq 0}} \frac{1}{n^2+m^2+k^2}e^{2\pi i(n,m,k) \cdot (r-...
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
80 views

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

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
32 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 $\lim a^{1/{x_n}}=0$.