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

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0
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1answer
16 views

Rewriting a particular sequence in respect to inverses

I'm having a large amount of difficulty on piecing together the intermediate algebra between the following formulas. $$ \frac{n^2 + 1}{2n^2 - 3} = \cdots = \frac {1 + \frac{1}{n ^ 2}}{2 - ...
0
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0answers
36 views

Show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$

To show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$ Let $u_r, u_{r+1}$ represent the $r^{th}$ and $(r + 1)^{th}$ terms of the expansion; then ...
0
votes
1answer
13 views

Regarding uniform and pointwise convergence

If a real sequence $(f_n)$ of functions converges to a function $f$ uniformly over a domain $D$ except at a a finite amount of points $x_1,\cdots,x_k$, but it happens that at each of these points, ...
1
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5answers
73 views

Convergence of $\sum_{n=1}^{\infty}(1-n\sin\frac{1}{n})$ [on hold]

Can someone help me to understand how to find out if this series absolutely convergent and regular converges: $$\sum_{n=1}^{\infty}(1-n\sin\tfrac{1}{n})$$
1
vote
1answer
24 views

Series convergence and Big O

I am trying to prove that if there exists $\theta \in \mathbb{R}$ such that $f(n) = \mathcal{O}(n^{\theta})$, then $\sum\limits_{n=1}^\infty \frac{f(n)}{n^s}$ converges. Intuitively it makes sense ...
0
votes
1answer
18 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 ...
4
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1answer
35 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
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2answers
46 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, ...
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4answers
41 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, ...
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0answers
36 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
35 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
25 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 ...
2
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0answers
24 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 ...
1
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2answers
34 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 ...
0
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3answers
23 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
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1answer
40 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!
0
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0answers
15 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 ...
8
votes
1answer
86 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
15 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, ...
1
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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.
1
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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.
0
votes
1answer
26 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
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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 ...
2
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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
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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
21 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
17 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 ...
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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 ...
2
votes
2answers
57 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 ...
0
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1answer
44 views

The set of infinite sequences with finitely many nonzero values is dense.

Could I get a proof to this lemma or a reference if a proof is too time consuming?
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 ...
0
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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
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1answer
22 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?
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 ...
0
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0answers
54 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 ...
0
votes
1answer
35 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 ...
3
votes
0answers
24 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k^n}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
0
votes
3answers
31 views

Determine the greatest value of $n$ for which $b > a$

Let $a_n$ and $b_n$ be two recursive sequences so that: $a_{n+1} = a_{n} + 2000$ where $a_{1} = 500$ and $b_{n+1} = \frac{b_{n}}{1,001}$ where $b_{1} = 50000$ Determine the greatest value of $n$ for ...
2
votes
1answer
28 views

Bounding infinite series derived from polygamma functions

Let $f(x) = 2 \psi^{(1)}(x+1) + x \psi^{(2)}(x+1) $ for $ x > 0 $, where $\psi^{(i)}(x)$ is the $i^{th}$ derivative of the digamma function $\psi(x)$. The goal is to prove that $ f(x) < ...
-1
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1answer
27 views

Find the point-wise limit of this sequence of function $\{f_n(x)\}$.

Consider the sequence of function $\{f_n(x)\}$ in $[0,1]$ where , $$f_n(x)=\begin{cases}0 & \text{ if } x=0\\n^2x & \text{ if } x\in [0,\frac{1}{n}]\\-n^2x+n^2 & \text{ if } x\in ...
1
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0answers
38 views

There exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(z)| = \infty$

Suppose $f$ has an Essential Singularity at $Z_0$. Then there exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(Z_n)| = \infty$ Here two cases arise If there exist a nbd ...
0
votes
2answers
25 views

Sums of converging limits

How can I prove the property that if the sequences, $(x)\rightarrow x' $ and $(y)\rightarrow y'$ then $(x) + (y)\rightarrow x'+y'$
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votes
2answers
59 views

Is $(-1)^{n!}$ convergent? [on hold]

I don't think I can use the alternating series test because of the factorial sign, but I don't know how else to solve this. can you please give any hints ?
0
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1answer
48 views

How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$

For reals $x \ge 3$, let $f(x)$ denote the function $f(x) = \frac{-x + x\sqrt{4x - 3}}{2}$. Now suppose that $a_1, a_2, \ldots, a_{2013}$ is a sequence of real numbers such that $a_1 > 3, a_{2013} ...
1
vote
0answers
29 views

Show that a sequence is between a range

I got this question in class which I'm having trouble proving I tried investigate the sequence a little bit but it doesn't seem like I'm doing the right think, some help? $ \frac{39}{e^2} \le ...
0
votes
1answer
27 views

Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$

I have this question which I'm having trouble solving, can I use some help? :) Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$: $ \sum_{n=1}^{\infty} ...
1
vote
3answers
54 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
0
votes
2answers
34 views

Error in a Maclaurin series

I'm having trouble figuring out what I have to do with this question. "Using Taylor's theorem, determine the largest positive real value $r$ for which we can guarantee that the Maclaurin polynomial ...
4
votes
2answers
56 views

Is it true that: $|a_{n+1} - L| < |a_{n} - L| \forall n \in \mathbb{N} \implies \lim \limits_{n \to \infty} a_{n} = L ?$

If $a_{n}$ is a sequence and $|a_{n+1} - L| < |a_{n} - L|, \forall n \in \mathbb{N} $, then clearly the sequence $s_{n} = |a_{n} - L|$ converges (it's decreasing and bounded by $0$). Does it ...
0
votes
1answer
47 views

Closed form for series involving harmonic numbers

Is there a closed form for this series values: $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(n)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^k \frac{1}{i^n} $$ and n is a positive integer. Thanks!