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

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3answers
47 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $(x_n)$ tending to $0$ such that $f(x_n)=0$ for all $n$, then $f(x)=0$ for all $x$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
2
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2answers
33 views

How does this manipulation of summations work?

I am reading some mathematics in which is the following algebraic manipulation. $$ \begin{align} \exp(x)\exp(y) & = \left(\sum_{n = 0}^\infty \frac{x^n}{n!}\right) \left(\sum_{m = 0}^\infty ...
2
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2answers
37 views

If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges?

If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges? I ...
2
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2answers
28 views

Determine the set of values of $x$ such that this series converge

Determine the set of values of $x$ such that this series converge: $$\sum^{\infty}_{n=1} \frac{e^n+1}{e^{2n}+n} x^n$$ My work: If $x\geq e$, we have $$\frac{e^n+1}{e^{2n}+n} x^n \geq ...
0
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1answer
33 views

Differentiate this power series

I am working on a problem which involves the differentiation of a power series. I know that that the following holds. Let $R$ be the radius of convergence of the power series $\sum_{n = 0}^\infty ...
2
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2answers
24 views

Proof that repeated sum equals binomial formula

Let $s, d$ be positive integers. Can you prove the following general formula for the repeated sum? I developed this problem on my own, but is it a well known result? $$\sum_{i_1 = 0}^s \sum_{i_2 = ...
0
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0answers
34 views

Doubt on the comparison test: can I still evaluate $\lim \limits_{n \to \infty} \frac{a_{n}}{b_{n}}$ if $b_{n}$ might be $0$ for some $n$?

Suppose that $(b_{n})_{n \in \mathbb{N}}$ is a sequence which is not identically equal to $0$ (but which may have elements equal to $0$). Suppose also that I know that $\sum b_{n}$ converges ...
0
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0answers
20 views

Showing sequence function is monotone

Is this sequence of functions monotone? $$f_{n+1}(x)=\varphi(x)f_n(x)+\frac{1}{\varphi(x) f_n(x)}, \forall x \in[0,+\infty)$$ Where $\varphi:[0,+\infty)\to \mathbb{R}$, $1/2\leq \varphi(x) <1$ ...
2
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3answers
46 views

Is the argument I used to evaluate the convergence of the series $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$ right?

If $a,b,c$ be real constants, analyze the convergence of $$\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$$ What I tried to: I compared the general term of my series to $\frac{1}{n}$: ...
1
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0answers
11 views

Special $\omega(n)$-sequence

Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$. The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a ...
2
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7answers
84 views

Tell if a sum is convergent $\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$

$$\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$$ I tried to solve this by saying that $$\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}$$ I then made two sums like this: $$\sum\limits_{n=1}^\infty ...
0
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2answers
31 views

Calcultaing function limit of a limit sequence

How do I even start? $$\lim_{x \to 1^+} \lim_{n \to \infty}{x^n \over x^n + 7}$$ I see that it should be $1$ but how do i prove it?
3
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1answer
39 views

Asymptotic ratio of two series

Assume $\{a_n\}$ and $\{b_n\}$ are two positive series such that $$\sum_{n}a_n=\sum_n b_n=1.$$ Assume also for all $n$, $\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k$ and $$\lim_{n\rightarrow ...
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0answers
22 views

How to quickly find $2^x \pmod N$ cycle

$N$ is a large composite number. If we don't have the factorization for $N$,how do we quickly find the order of $2$ in $(\mathbb{Z}_N, \cdot_N)$? Example. Suppose $N$ is ...
1
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1answer
34 views

Series in hyperbolic sines.

I was looking into a problem and I arrived to something in which I want to expand some function $\varphi(x)$ in series of hyperbolic sines, something like: ...
-1
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1answer
26 views

Uniform convergence of series $\sum_{n=0}^\infty \left(\frac{z-1}{z+1}\right)^n $

I'm having trouble with uniform convergence. I need to prove that $$\sum_{n=0}^\infty \left(\frac{z-1}{z+1}\right)^n $$ converges locally uniformly in the half-plane $Re z >0$ and find its sum. ...
2
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1answer
75 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, ...
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1answer
26 views

Determine the sum of the following series. [on hold]

Determine the sum of the following series: $$\sum_{n=1}^\infty\frac{(-3)^{n-1}}{n^5}$$
0
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5answers
52 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.
0
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3answers
25 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
21 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
38 views

Classification of Series $(-1)^n$ [on hold]

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
40 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
26 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
383 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
9 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
18 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
68 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 $$ ...
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1answer
21 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
30 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
20 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
23 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
votes
1answer
17 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, ...
0
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5answers
97 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: $$\sum_{n=1}^{\infty}(1-n\sin\tfrac{1}{n})$$ It's a sequence. It's not a function.
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
24 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
52 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
votes
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
42 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
39 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
votes
0answers
16 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
94 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
16 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, ...