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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2answers
33 views

Two sequences, one of them bounds difference of the other and converges to $0$. Show that the other sequence converges.

That is, let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be sequences such that $b_n \overset{n \to \infty}{\to} 0$ and for all $k \in \mathbb{N}$ and $l \geq k$, $$|a_l - a_k| < b_k\text{.}$...
1
vote
0answers
51 views

Finding hte value of $1/(1^2)+1/(2^2)+1/(3^2)+\dots +1/(n^2)$ [duplicate]

Sorry, my English isn't quite well. Here is the problem. When $n$ moves toward infinity $$ 1/(1^2)+1/(2^2)+1/(3^2)+\dots +1/(n^2)=? $$ Any ideas?
0
votes
0answers
65 views

Formula for $\pi$ using primes

In one of his videos (https://www.youtube.com/watch?v=HrRMnzANHHs), Matt Parker introduces the following formula for $\pi$ using primes: $$\left(1-\frac{1}{3}\right)\cdot\left(1+\frac{1}{5}\right)\...
0
votes
1answer
16 views

Cauchy with terms of differing signs after some term $\implies$ convergence

Let $(x_n)_{n=1}^{\infty}$ be Cauchy. For each natural number $N$ (natural numbers starting at $1$), suppose there is an $n_1 \geq N$ and $n_2 \geq N$ such that $x_{n_1} < 0$ and $x_{n_2} > 0$. ...
0
votes
2answers
91 views

Find $\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$

I have problem with finding the following limit. I suspect that it should be easy, but really I don't have a clue. $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} $$
0
votes
0answers
10 views

Composition of limits of functions | Switching limits of function

I have a question which I am having some trouble with. I have a double indexed sequence of stochastic processes (martingales in fact), denoted $X_{m,n}(t)$. Now I can prove that $\underset{m \...
0
votes
1answer
27 views

Nested Hypergeometric series

Is it possible to express the following series as a hypergeometric function: $$\sum_{n=0}^\infty (a)_n \sum_{j_1+j_2+\cdots+j_k=n} \frac{1}{(b)_{j_1} (b)_{j_2}\cdots (b)_{j_k}} z^n $$ where $(a)_n, (...
0
votes
1answer
27 views

Is it true that $(\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw$?

Let $v,w$ be vector fields and let $\Phi_t$ be the flow generated by $v$, that is $\frac{d}{dt}\big|_0\Phi_t(x)=v(x)$. The Lie derivative of $w$ in direction of $v$ is usually defined as $\mathcal ...
8
votes
2answers
243 views

Limit of the sequence $a_{n+1}=\frac{1}{2} (a_n+\sqrt{\frac{a_n^2+b_n^2}{2}})$ - can't recognize the pattern

Consider the sequence: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\frac{1}{2} \left(a_n+\sqrt{\frac{a_n^2+b_n^2}{2}} \right),~b_{n+1}=\frac{1}{2} \left(b_n+\sqrt{\frac{a_n^2+b_n^2}{2}}\right)$$ $$\lim_{n \to \...
0
votes
0answers
50 views

Seeking general formula for Euler Sum $\sum\limits _{ n=1 }^{ \infty }{ \frac {{\left({H}_{n}\right)}^{p}}{{n}^{q}}}$ with $q$ even and $p$ odd

I am wondering if there is a formula for $\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left( { H }_{ n } \right) }^{ p } }{ { n }^{ q } } }$ with 'q' being an EVEN positive integer and 'p' ...
1
vote
1answer
37 views

Proof of convergence of an infinite sequence

A question that I tried to prove is as follows. "Consider the following sequence defined recursively by $a_1=\sqrt{a}$ and $a_{n+1}=\sqrt{a+a_n}$, where $a>2$. (The first few terms are: $\sqrt{a}, ...
3
votes
1answer
41 views

$f \in C [0,1]$ , $\lim _{x \to 0+} f(x)/x$ exists finitely , $\lim_{n \to \infty} n\Big(n \int_0^1 f(x^n)dx-\int_0^1 \dfrac {f(x)}x dx\Big)=$? [closed]

Let $f:[0,1] \to \mathbb R$ be a continuous function such that $\lim _{x \to 0+} \dfrac {f(x)}x$ exists finitely . Then does the limit $\lim_{n \to \infty} n\Big(n \int_0^1 f(x^n)dx-\int_0^1 \dfrac {f(...
0
votes
1answer
74 views

Find the $m$ such $a_{n+1}=a^5_{n}+487$ [closed]

Let $\{a_{n}\}$ be a sequence of positive integers, and suppose $a_{0}=m$. Further, $\{a_{n}\}$ satisfies $$a_{n+1}=a^5_{n}+487.$$ Find $m$ so that this sequence consists of square numbers for as long ...
2
votes
1answer
34 views

Sum of cosines with a multiplicative factor in the angle and different interval

I have found the following formula for the sum of cosines in both here and here. \begin{align} \sum^n_{l=1} \cos \left(\frac{2 \pi l}{n}\right) = 0 \end{align} I would like to know what the sum ...
1
vote
1answer
33 views

Pointwise limit of $\sum{\frac{1}{n} \left(\frac{2z-i}{2+iz}\right)^n}$?

I am trying to find the set $\Omega$ where the series $$ \sum_{n \geq 1} \frac{1}{n} \left(\frac{2z-i}{2+iz}\right)^n $$ exhibits pointwise convergence. I have thought of several approaches: ...
3
votes
1answer
49 views

Show that being a sequential topological space is preserved under homeomorphism

Definition 1: $A \subset (X, \mathcal{T})$ is sequentially closed if the limit of all convergent sequence $(x_n)$ in $A$ is in $A$. Definition 2: $(X, \mathcal{T})$ is a sequential topological ...
2
votes
1answer
37 views

A BBP-type series

The BBP-type series $$ \frac{\pi}{2} \, \left( \frac{\alpha^{2}}{5} \right)^{\frac{1}{4}} = \sum_{n=0}^{\infty} \left[ \frac{1}{10 n + 1} + \frac{\alpha}{10 n + 3} - \frac{\alpha}{10 n + 7} - \...
4
votes
2answers
76 views

The sequence $(a_n)$ is given as $a_1=1, a_{2n} = a_n - 1, a_{2n+1} = a_n + 1$. $a_{2015}=$?

The sequence $(a_n)$ is given as $a_1=1, a_{2n} = a_n - 1, a_{2n+1} = a_n + 1$. What's the value of $a_{2015}$ Correct answer should be $a_{2015} = 9$. How? thing that came to mind was to see what $...
0
votes
0answers
28 views

How to Solve a Function Given Some of its Solutions

Suppose you have a function that defines a series. And suppose you know Some (not all) of the elements of that series. For example, you know your function is n/J, where n is for all positive integers ...
0
votes
2answers
52 views

Triangular numbers and pascal's trangle

The following are the triangular numbers. rank = 1 2 3 4 5 6 term = 1 3 6 10 15 21 A rule for triangular numbers is: ...
2
votes
2answers
65 views

Trouble with proving uniform convergence

prove : $$\sum_{n=1}^{\infty } \dfrac{nx}{e^{nx}}$$ converge for each x $\in R$ , converge uniformly in $[\alpha,\infty)$ for $\alpha >0$ and doesn't converger uniformly at $[0,\infty)$. Having ...
1
vote
1answer
48 views

Reverse and forward doubling identity in Fibonacci sequence $\text{mod 9}$

Fibonacci sequence ($\mathbb{F}$) has a repeating cycle known as Pisano number $\pi\text{(x)}$ , when $mod \text{ x}$ is applied upon the sequence. Length of the cycles can be found from: http://oeis....
2
votes
2answers
173 views

Does there exist prime number of the form $0^0+1^1+2^2+3^3+4^4+…$ after the trivial one $2$?

I am interested with prime numbers of the form $0^0+1^1+2^2+3^3+4^4+....(n-1)^{n-1}+ n^n$ (where we take $0^0=1$). I've checked $n$ up to $250$, and I found that numbers of such form are very very ...
2
votes
2answers
32 views

What is the sum of the first $17$ terms of an arithmetic sequence if $a_9=35$?

What is the sum of the first $17$ terms of an arithmetic sequence if $a_9=35$? This is what I did: $a_9=a_1+8d=35$ $S_{17}=\frac{17}{2}(a_1+a_{17})=\frac{17}{2}(a_1+a_1+16d)=\frac{17}{2}(2a_1+16d)=\...
3
votes
4answers
67 views

write an explicit formula for the sum $\sum_{i=1}^n {(3i+1)}$

I've been shown that : $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ Now I need to write an explicit formula for the sum: $$\sum_{i=1}^n {(3i+1)}$$ I've come up with an answer that is: $$\sum_{i=1}^n {(3i+...
1
vote
1answer
41 views

finding the limit of a sequence including sigma symbol

I have a sequence : $$\sum_{k=1}^n e^{\frac{k}{n^2}}\times \ln(k+\frac{1}{k})$$ I have to find the limit of this sequence , I tried to surround it ,but this not the correct way.
7
votes
1answer
64 views

Prove that there exists a sequence $(x_n)$ such that $\sum_n a_n x_n$ diverges

So, here's a nice little result that I deduced using the closed graph theorem from functional analysis, but I'm wondering if there's a more elementary approach: Fact: Let $(a_n)$ be a sequence ...
1
vote
0answers
34 views

conditionally convergence [closed]

It is given that the series $\sum_{n=1}^{\infty}a_n$ is convergent, but not absolutely convergent and $\sum_{n=1}^{\infty}a_n=0$. Denote by $s_k$ the partial sum $\sum_{n=1}^{k}a_n$ for $k=1,2,...$ ...
3
votes
1answer
51 views

Does iterating $x \cdot \sin(\frac 1 x) + x$ near $0$ approach $0$?

Let $f^1(x) := x\,\sin(\frac{1}{x})+x$ and define $f^N (x):= f(f^{N-1}(x))$ for $N\in \mathbb{Z},\ N>1$. For which $x \in \mathbb{R}$ does $\lim_{N\rightarrow\infty}{f^N(x)}=0?$ Clearly, for $x\...
4
votes
6answers
119 views

Prove that $ 1+2q+3q^2+…+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $

Prove: $$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$ Hypothesis: $$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$ Proof: $$ P1 | F(x) = \frac{1-(...
4
votes
1answer
310 views

Prove $\int_{0}^{\infty}{\phi^3e^{4x}-\phi^4e^{3x}-\phi^3e^{2x}+e^x+2\over (\phi e^x)^5-1}\cdot2xdx=\left({\pi\over 5}\right)^2$

Prove the following equation, given that $\phi$ stands for the golden ratio: $$I=\int_{0}^{\infty}{\phi^3e^{4x}-\phi^4e^{3x}-\phi^3e^{2x}+e^x+2\over (\phi e^x)^5-1}\cdot2xdx=\color{blue}{\left({\...
2
votes
1answer
63 views

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n ...
13
votes
5answers
2k views

Are we guaranteed that the harmonic series minus infinite random terms always converge?

Consider the known harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ and modify it as follows $$\sum_{n=1}^\infty a_n\frac{1}{n}$$ where $$a_n \sim \operatorname{Bern} \left({\frac{1}{2}}\right)$$ i.e. ...
1
vote
1answer
65 views

Summing power series $\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$

Lets have series $$\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$$ Obviously, its convergence radius is 1. I should sum it, but don't know what's up with the double factorial. There is a hint in the ...
3
votes
1answer
86 views

Limit and rate of convergence of the sequence $a_{n+1}=\frac{a_n^2+b_n^2}{a_n+b_n},~~b_{n+1}=\frac{a_n+b_n}{2}$

Define the sequence the following way for some $x,y \geq 0$: $$a_0=x,~~~~~~~b_0=y$$ $$a_{n+1}=\frac{a_n^2+b_n^2}{a_n+b_n},~~~~~~b_{n+1}=\frac{a_n+b_n}{2}$$ Obviously: $$a_n \geq b_n,~~~~n \geq 1$$ ...
3
votes
2answers
133 views

Prove this series does not uniformly converge

$$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ It is easy to show that it absolutely converges. But what about uniform convergence? With M-test: $$|| f_n|| = \sup (| n^2 \sin \frac{x}{n^4}|) \leq \...
0
votes
1answer
54 views

If $\limsup(na_n) = 1$, then $\sum\limits_{n=1}^{\infty} a_n$ diverges

Let $a_n$ be a sequence of positive numbers. Suppose $\limsup(na_n) = 1$. Does this mean $\sum a_n$ diverges? I have only concluded this if the limit superior is in fact the limit of the sequence. ...
1
vote
1answer
49 views

Does the series $\frac{1}{(\log{n})^4}$ converge?

I have used comparison test to show it diverges: $$\frac{1}{n}<\frac{1}{(\log{n})^4}$$ But is this even right?
0
votes
0answers
23 views

Closed unit ball in $\ell^p$ is compact

I'm curious as to whether the closed unit ball in $\ell^p$ is compact for $1 \leq p \leq \infty$, with respect to the $p$-norm, and the $\sup$ norm?
1
vote
2answers
49 views

Evaluating the series in simple way

Evaluate the series $$\sum_{n=1}^{\infty} \frac{2n +1}{n^2(n+1)^2}$$ I know to evaluate such series, I need to evaluate the limiting value when $n\to\infty$ of the partial sums of the series, but I ...
1
vote
1answer
35 views

Sequences of 0s and 1s are compact

Let $X$ be the space of sequences $x = (x_1, x_2, ..., x_n,...)$ such that $x_i = 0$ or $x_i=1$, equipped with the metric $$d(x,y) = \sum_{n=1}^{\infty} \frac{1}{2^n} \left| x_n - y_n \right|.$$ Prove ...
3
votes
4answers
53 views

$\lim\limits_{n\to \infty}\sum_{k=1}^{\infty}2^{-k}\sin(k/n)=0$

I want to show that $$\lim\limits_{n\to \infty}\sum_{k=1}^{\infty}2^{-k}\sin(k/n)=0$$ I first thought if I can change the order of limit, it can be easy to show that. But I found that there ...
4
votes
0answers
211 views

Research: Looking for a sequence that produce variation's of Pascal's triangle

Prologue I am an undergraduate so if my terminology or approach seem inappropriate/confusing please explain in the comments. I created a notation where $$F(0 \rightarrow n,x) = [\hspace{1mm}F(0 ,...
0
votes
1answer
32 views

Solving this generating function to find the $n$th term in the sequence

I have been given the generating function $$f(x) = \frac{x^2+x+1}{1-x^7},$$ and I need to solve for a closed form of the $n$th term of the sequence g generated by this function. I have been trying to ...
1
vote
0answers
42 views

Characteristic polynomial and Generating Function for recurrence relation of an integer sequence

Given an integer sequence, such that for $n > 2$, $a_n$ = greatest number of the form gpf $(a_{n-1})k$ + spf $(a_{n-2}) \le n^{2}$. Where gpf denotes the Greatest Prime Factor and spf the Smallest ...
2
votes
0answers
60 views

Evaluating $ \sum_{n=0}^\infty \frac{1}{1+n(1-e^{an})} $

I don't know if a closed form exists, but if it does, I'd like to know: $$ \sum_{n=0}^\infty \frac{1}{1+n(1-e^{an})} $$ for $a>0$ - assume that the value of $a$ is such that the denominator is ...
0
votes
0answers
19 views

Convergance of the average of a convergant complex sequence

So this is in exercise 3.14 (Neat!) in Baby Rudin, which I have found quite a simple and obvious proof for, however when I checked the answers the proofs I found were quite complicated so now I am a ...