For questions about recurrence relations, convergence tests, and identifying sequences

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0
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1answer
14 views

Laurent series: how to join the 2 sums for $f(z)= \frac{1}{(z-1)(z+1)}$ about z = 1 for $0 < |z − 1| < 2$

We are to find the Laurent series for f(z) about $z = 1$ for $0 < |z − 1| < 2$: $f(z)= \frac{1}{(z-1)(z+1)}$ Assumptions: $|\frac{z−2}{1}| < 1 ⇔ |z − 1| < 2$ For $\frac{1}{(z-1)}$ we ...
1
vote
2answers
36 views

Finding the limit of a recursively defined sequence (recurrence relation). Specifically and generally

Whilst reading Goldrei's Classic Set Theory, I have come across a recursively defined sequence $a_0=0, a_1=1, a_n=\frac{1}{2}\left(a_{n-1} + a_{n-2} \right) $ The first few terms of which are: $0, ...
4
votes
1answer
33 views

Compute finite series

The problem is to count the sum of the finite series $$\sum_{k=0}^{k_0} \frac{a_k}{b_k}$$ I need to count this series in binary with some precision, that would output $n$ correct binary digits after ...
1
vote
0answers
14 views

Complementary Golay sequences and sum of their autocorrelation function

Golay complementary sequences are aperiodic sequences made up of +1 and -1 that have nice property which is that their autocorrelation that sum up as korneckr delta function. Example $G_{a4}=(+1, +1, ...
0
votes
4answers
23 views

Proving a recursive sequence is bounded

I'm proving that the limit of the following recursive sequence is $\dfrac{10}{9}$: $$s_0=1,\,s_n=s_{n-1}+\frac{1}{10^n}\quad\text{for }n\ge1$$ Showing that the sequence is monotonic was easy enough, ...
1
vote
1answer
32 views

Any suggestions to decide whether $\sum_{n=1}^{\infty} \frac{\sqrt{2n-1} \ln (4n+1)}{n(n+1)}$ converges or not?

First, I verified if the general term $\frac{\sqrt{2n-1} \ln (4n+1)}{n(n+1)}$ tends to $0$, and it does: $$\lim \limits_{n \to \infty} \frac{\sqrt{2n-1}}{n} \frac{\ln(4n+1)}{n+1} = 0$$ Which other ...
2
votes
1answer
34 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
0
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0answers
19 views

Let $\{x_n\}$ be a sequence of positive reals, show that $\limsup\sqrt[n] {x_n}\leq\limsup \frac{x_{n+1}}{x_n}$.

Let $\{x_n\}$ be a sequence of positive reals, show that $\limsup\sqrt[n] {x_n}\leq\limsup \frac{x_{n+1}}{x_n}$. This come from a problem set, in which $\limsup{\sum^{n}_{i=1}\frac ...
4
votes
1answer
94 views

The infinite series $\sum_i a_i \prod_{j=0}^{i - 1}(1 - a_j)$ has sum equal to $1$

Suppose we have an infinite sequence of constants $\{ a_i \}_{i=0}^{\infty}$, where $0 < a_i < 1$ for every $i$, and we define $b_i \equiv a_i \prod_{j=0}^{i - 1}(1 - a_j)$. How do we prove ...
1
vote
1answer
37 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
0
votes
0answers
13 views

Given $z_n$ = {$ x_{1},y_1,x_2,y_2,x_3,y_3 …$} Prove that $z_n$ is convergent if $x_n$ and $y_n$ both converge to same limit [duplicate]

I f $x_n$ and $y_n$ be the two sequences such that $z_n$ = {$ x_{1},y_1,x_2,y_2,x_3,y_3 ...$} Prove that $z_n$ is convergent if $x_n$ and $y_n$ both converge to same limit ATTEMPT Let us take that ...
1
vote
2answers
33 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
1
vote
2answers
432 views

Difference between divergent series and series with no limit?

Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.
2
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0answers
20 views

Convergence of a series of integrals

Definition of problem I would like to find the large $N$ behaviour of a summation series and specifically the conditions under which it converges. I have found one condition, but I think it might be ...
0
votes
3answers
22 views

Show convergence of 1/cosh series

Sorry if my english is not correct. Feel free to edit and ask questions. I need to test the following series on convergence: a) $$ \sum_{n=0}^{\infty}\frac { sinh(n) }{ e^n } $$ and b) $$ ...
1
vote
1answer
37 views

Exact value of a sum involving harmonic numbers

Could somebody tell me the exact value of this series? $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n} $$ Thanks!
8
votes
1answer
83 views

Limits, Taylor expansion

Find the limit: $$ \lim_{x\to\infty} \frac{\displaystyle \sum\limits_{i = 0}^\infty \frac{x^{in}}{(in)!}}{\displaystyle\sum\limits_{j = 0}^\infty \frac{x^{jm}}{(jm)!}} $$ for $n$, $m$ natural ...
1
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0answers
20 views

Infinite sums over integral of triple associated Legendre polynomials

I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really ...
9
votes
2answers
360 views

Convergence of the sequence $\frac{1}{n\sin(n)}$

Does the sequence $$\frac{1}{n\sin(n)}$$ converge to $0$ or not? If not, what's the upper limit?
5
votes
1answer
1k views

Continuous Functions and Cauchy Sequences

We know that if a function $f: A \mapsto \mathbb{R}$, $A \subseteq \mathbb{R}$, is uniformly continuous on $A$ then, if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is also a Cauchy sequence. ...
0
votes
0answers
11 views

How to use finite differences to find approximate functions for a set of data?

Given the data: x: 0 2 4 6 8 10 12 14 16 y: 5 -8 -11 -9 4 23 52 89 131 How would you use finite differences to find the approximate function to model this data? There is no common ...
0
votes
1answer
25 views

Interval of convergence of power series?

If the power series is: $$ \sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n+1}} $$ and I've found the interval to be $$ -1 < x < 1 $$ then would the answer $$ (-1, 1) $$ work? some other options ...
1
vote
0answers
33 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
0
votes
1answer
44 views
1
vote
1answer
28 views

Power series representation?

The function to represent as a power series is: $$ \frac {10} {(x-10)^2} $$ Any help is, as always, appreciated.
2
votes
3answers
29 views

(Simple question) Radius of covergence of power series?

For the power series: $$ \sum\limits_{n=1}^{\infty}\frac{(x-1)^n}{2^n} $$ Would radius of convergence be $$ x = 1 $$ ?
2
votes
2answers
56 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
2
votes
1answer
41 views

What is the center of power series?

The power series is: $$ \sum\limits_{n=1}^{\infty}\frac{(x+4)^n}{n+1} $$ Any help appreciated!
0
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0answers
19 views

$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$ convergence

Does $$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$$ converge? Domain of this integrand is $x \in \mathbb{R} : x\neq 0, ...
1
vote
1answer
14 views

Sequence from generating function with integral

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$ Since I am not much acquainted with integrals any help ...
0
votes
1answer
45 views

Integrating $\frac{1}{1+z}$ along a path to derive the Maclaurin series for $\mathrm{Log}(1+z)$

Integrate the Maclaurin series for$\frac{1}{1+z}$ along a path, inside the circle of convergence, going from $z'=0$ to $z'=z$ and show that $$Log(z+1)=\sum_{i=1}^\infty (-1)^{n+1}\frac{z^n}{n}, ...
1
vote
2answers
35 views

Let $x_n$ be sequence converging to $0$ . What can you say about sequence $(x_n)^{n}$

Let $x_n$ be sequence converging to $0$ .What can you say about sequence $(x_n)^{n}$ ATTEMPT $|x_n|<\epsilon^{1/n}$ for all $n \geq$ m implies $ |x_n|^{n} < \epsilon $. Thus new sequence is ...
1
vote
1answer
71 views

If $x_{n}$ and $x_{n}y_{n}$ are bounded, does it follow that $y_{n}$ is bounded? [closed]

If $x_{n}$ and $x_{n}y_{n}$ are bounded, does it follow that $y_{n}$ is bounded? Attempt Let |$x_{n}| \leq C$ and |$x_{n}y_{n}| \leq C'$, then |$x_{n}y_{n}|$ $\leq$ $ |y_{n}|$ $\leq C'/C$. If ...
2
votes
1answer
234 views

Permuting the terms of a sequence does not affect its convergence

Let $x_n$ be a sequence such that $x_n \rightarrow 0$. Let $\sigma\colon\mathbb N \rightarrow \mathbb N$ be a bijection. Define a new sequence $y_n:= x_ {\sigma (n)} $. Show that $ y_n \rightarrow 0 ...
-2
votes
1answer
37 views

Understanding proof that real sequence is Cauchy iff it is convergent [closed]

I am having trouble understanding what is motive and idea behnd th proof given here
2
votes
2answers
32 views

Continuity and differentiability of $f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!(x+n)}$

Given the series: $$\sum_{n \ge 0} \frac{(-1)^n}{n!(x+n)}$$ Let $f_n(x)$ denote its general term. Let $f(x)$ denote its sum (when exists). The question asks to: $i)$ Find the domain $\mathbb D$ on ...
4
votes
0answers
125 views

Is the infinite sum $\sum_{s=2}^\infty \frac{\zeta(s)}{s!}$ known? If so, what is its value?

I recently ran into this infinite sum: $$\sum_{s=2}^\infty \frac{\zeta(s)}{s!}$$ and have tried to solve it to no avail. Any references, solutions, or general advice would be greatly appreciated.
0
votes
1answer
42 views

Proving a series from zero to infinity is half of a series from minus infinity to infinity?

I want to prove that $$\sum_{n=-\infty}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$ is equivalent to $$\sum_{n=0}^{\infty} \frac{2(-1)^n}{(2n+1)^3}$$ I have played around with it and I know that it is correct ...
-4
votes
1answer
21 views

Absolutely convergent, conditionally convergent or divergent [on hold]

I have this question: $$\sum_{n=1}^\infty \frac{\cos\left(\frac{n\pi}{12}\right)}{n\sqrt n} $$ How do I figure out if it's absolutely convergent, conditionally convergent or divergent?
0
votes
0answers
53 views

Finding the limit of this specific series

So, I have to calculate: $$\lim _{ n\to\infty } \prod_{k=2}^{n} \Big(2-\sqrt[k]{2}\Big)$$ So far I managed to get to: $$\lim _{ n\to\infty } \sum_{k=2}^{n}\Big(1-\sqrt[k]{2}\Big)$$ Any help will ...
1
vote
1answer
101 views

series $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+…$ [closed]

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$ $\frac{\text{nth term}}{(n-1)\text{th term}}=\frac{n-1}{n}=1-\frac{1}{n}$ As $n\rightarrow \infty, \frac{\text{nth term}}{(n-1)\text{th ...
6
votes
2answers
1k views

Why is the ratio test for $L=1$ inconclusive?

One of the often used tests for convergence ($L\lt 1$) and divergence ($L\gt 1$) of an infinite series is the ratio test. The idea behind it, why it works is the geometric series which dominates (or ...
3
votes
2answers
77 views

If $\frac{a_{n+1}}{a_{n}} \nearrow 1$ when $n \to \infty$, does $\sum_{n=1}^{\infty} a_{n}$ converge?

Suppose $(a_{n})$ is a sequence which satisfies $a_{n} > 0, \forall n \in \mathbb{N}$. The ratio test states that if $\frac{a_{n+1}}{a_{n}} \to L < 1$ when $n \to \infty$, then the series ...
2
votes
1answer
98 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
2
votes
3answers
47 views

Determine a closed form for this sequence

Every year, 38 % of the amount of fish in a pond die. The 1st of May 2011 there were 5200 fish in the pond. Every year after May 1st 2011, 1900 new fish are added to the pond. Let $a_n$ be the amount ...
5
votes
0answers
138 views
+50

The recurrence $a_k(n) = \sum_{0\leqslant j<n} a_{k-1}(n+j)$

I'm trying to find a closed form expression to the following recurrence relation: \begin{align} &a_k(0) = 0, \quad \forall k\geqslant 1; \\ &a_k(1) = 1, \quad \forall k\geqslant 1; \\ ...
2
votes
2answers
332 views

Series convergence - Gauss test

How do I prove that $$\sum_{n=1}^\infty\left(\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot(2n)}\right)^{k}$$ converges for $k>2$ using Gauss test?
0
votes
1answer
34 views

Example of a convergent series for which integral test fails?

Is there example of a convergent series for which integral test fails or can not be applied? Just wondering if integral test is the silver bullet of convergence tests, or are there any series that any ...
0
votes
1answer
44 views

Prove/disprove statement about positive sequence which tends to infinity

Let $\{a_n\}$ be a positive sequence. I have to verify the following statements: If $\lim_{n\to\infty}a_n=\infty$ then $\sqrt[n]{a_n}>1+\frac{1}{n}$, for all but a finite number of $n$ If ...
-1
votes
2answers
74 views

sumation question

What is $\sum\frac{1}{1+2n+n^2}$ from 1 to infinity? I did $\sum\frac{1}{1+2n+n^2}=\sum 1+\frac{1}{2n}+\frac{1}{n^2}=\infty+\infty+\pi^2/6=\infty$ but apparently its wrong. But we saw in class that ...