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

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1answer
44 views

$1-1+1-1+1-1+\cdots = \frac 12$ proof?

Note: I claim no credit for this "proof". A friend came up with it and I thought it was pretty cool. Let's say you wanted to prove that $$1-1+1-1+1-1+ \cdots = \frac 12$$ Well, as mentioned before, ...
-1
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1answer
17 views

Determine the sum of the following series.

Determine the sum of the following series: $$\sum_{n=1}^\infty\frac{(-3)^{n-1}}{n^5}$$
0
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5answers
25 views

Define a sequence {$\ x_n$} recursively, show it is strictly decreasing

Define a sequence {$\ x_n$} recursively by $$ x_{n+1} = \sqrt{2 x_n -1}, \ and \ x_0=a \ where \ a>1 $$ Prove that {$\ x_n$} is strictly decreasing. I'm not sure where to start.
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3answers
22 views

Find the sequence of partial sums for the series $a_n = (-1)^n$ Does this series converge?

Find the sequence of partial sums for the series $$ \sum_{n=0}^\infty (-1)^n = 1 -1 + 1 -1 + 1 - \cdots$$ Does this series converge ? My answer is that the sequence $= 0.5 + 0.5(-1)^n$. This makes a ...
0
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1answer
20 views

Calculate the limit of the sequence by applying the limit laws?

I'm not sure how to approach this problem since its a bit different to the usual questions about calculating limits .
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3answers
31 views

Classification of Series $(-1)^n$

Does it converge or diverge or we can't tell? $$∑_{n=1}^{\infty}(-1)^n$$ Or is there simply no concrete answer? Thanks in advance.
2
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1answer
36 views

What is the Taylor series of $e^x$ centred at $3$?

$$ \sum_{k=0}^n \frac{e^3}{n!}(x-3)^n $$ This is my answer - is it correct?
2
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0answers
19 views

How to find coefficients in a power of a trig series?

Suppose $m$ is a positive integer, and we know that the following identity holds for all $0<x<2\pi$: ...
4
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2answers
276 views

Does this sum converge, is my solution good?

$$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7}+1)^{1/2}} $$ I would say that it doesn't converge, cause I would write this as: $$ $$ $$ \frac{\sin(n)^{7}}{(n^{7})^{1/2}} $$ when $$ \lim_{n\to ...
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0answers
8 views

Problems understanding Autocorrelation/Autokovariance

I am having some trouble understanding the concept of autokovariance/autokorrelation with a timelag l. From how i understand it, it is the kovariance/korellation a series has, with a timelagged ...
0
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1answer
17 views

Bernoulli odd numbers are 0 $B_{2n+1}=0,\;n>0$

I left the Maclaurin expansion of the function $f(x)=x/(e^x-1)$ $$\frac{x}{e^x-1}=\sum_{k=1}^{\infty} B_k\frac{x^k}{k!}=B_0\frac{x^0}{0!}+B_1\frac{x^1}{1!}+\sum_{k=2}^{\infty}B_k\frac{x^k}{k!}$$ $$ ...
2
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1answer
56 views

infinity series of Riemann zeta function at odd integers

Properties of Riemann zeta function at odd and even integers diverge dramatically, which can be proved by many evidences. I once found an infinity series in wikipedia, it reads $$ ...
-2
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1answer
20 views

For which real x does the following series converge [on hold]

For which $x\in\mathbb R$ does the series $\Sigma\ x^{n!}$ converge?
2
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1answer
29 views

How to determine the generating function?

So I have $$\overset{*}{F} = \overset{*}{F}_{n-1} + \overset{*}{F}_{n-2} + g(n)$$ where $\overset{*}{F}$ is NOT a Fibonacci number for $n \geq 2$. $g(n)$ is any function $g: \mathbb{N} \to ...
0
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1answer
16 views

Solving a series in the proof of the expectation of the binomial distribution

I am studying the expectations and variances of the most common distributions. For the binomial distribution the mean is equal to $np$. Considering $p$ and $q$ independent variables and ...
0
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2answers
22 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
39 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
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1answer
16 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
79 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
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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
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1answer
22 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
38 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
50 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, ...
0
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4answers
42 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
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1answer
40 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
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1answer
37 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
28 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
25 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
38 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
24 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
41 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
90 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
29 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
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1answer
27 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
38 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
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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 ...
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 ...
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
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2answers
33 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 ...
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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
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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
votes
0answers
55 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 ...