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Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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0answers
5 views

Proof: $\exists z\in \mathbb{C}$ such that every sub-sequence of $(z_n)_n$ has a convergent sub-sequence with the limit $z$.

$\exists z\in \mathbb{C}$ such that every sub-sequence of $(z_n)_n$ has a convergent sub-sequence with the limit $z$. My attempt: Sei $z_{{n_k}_k}$ be a sub-sub-sequence of $(z_n)_n$. It is $n_{k_k}\...
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0answers
13 views

Sufficient condition for positive alternating series with binomial coefficient

I have the following function: \begin{equation} F(\tau) = \sum_{k=0}^{\tau}\frac{C_k^\tau(-1)^k \left[ {}_2F_1\left(-2/\alpha,k;1-2/\alpha;-\gamma\right)-1\right]}{\left[1+L({}_2F_1\left(-2/\alpha,k;1-...
6
votes
1answer
81 views

Show that the $n$-th Fibonacci number is given by $\frac{\cosh na}{\cosh a}$ or $\frac{\sinh na}{\cosh a}$, where $\sinh a=1/2$

This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis. I request to help find the solution. ...
3
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0answers
35 views

Products of trig functions and the Thue–Morse sequence

I was studying transformations of finite products of trig functions into sums, and empirically observed that the following curious identity appears to hold for all non-negative integer $m$: $$\prod_{...
4
votes
2answers
111 views

If $\sum a_n$ converges , then $a_n<1/n$ a.e?

If $\sum_{n=1}^{\infty}a_n<\infty$ is a Positive convergent series, does the following limit hold? $$\lim_{n\to\infty}\frac{\mathrm{Card}\{1\leq k\leq n , a_k\geq\frac{1}{k}\}}{n}=0$$ I know of a ...
0
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1answer
47 views

Recurrence $f_{n+2}=af_{n+1}+bf_n$

Solve the recurrence $$f_{n+2}=af_{n+1}+bf_n\qquad n\in\Bbb N_0\tag{1}$$ Where $a,b>0$ and $f_0,f_1$ are given. I know that if $$F_{n+1}=c_nF_n+d_n$$ then $$F_n=F_0\prod_{k=0}^{n-1}c_k+\sum_{m=0}...
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3answers
32 views

Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null sequence (a sequence tending to $0$)

Let $z\in \mathbb{C}$, $(z_n)_{n\geq 1} \subset \mathbb{C}$ and $(w_n)_{n\geq 1}$(a null sequence) be sequences. Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null ...
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1answer
72 views

Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ [duplicate]

I want to prove that $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ There is a solution of the sum provided in my text book. There the expansion of $(1+\frac{1}{n})^n$ is like below: $(1+\frac{1}{n})...
5
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1answer
70 views

Show $\lim_{n\to\infty} na_n = 2$ [duplicate]

Suppose $a_n$ is real sequence which satisfies $$ a_1>0, \quad a_{n+1}=\ln(a_n+1) \quad (n\geq1)$$ How can I evaluate $$\lim_{n\to\infty}na_n$$? I just know $$\lim_{n\to\infty} a_n=0$$ But I don'...
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0answers
11 views

Notation for subsequences: Meaning of “primary”/“top” index and how to write unions?

Part 1: For some sequence $\{x_n\} = \{x_1, x_2, ...\}$, if I write a subsequence as $\{x_{n_k}\} = \{x_{n_1}, x_{n_2}, ...\}$ where $n_1 < n_2 < ...$, what is the meaning of the index $n$ in ...
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0answers
15 views

Find the equivalent of a sequence [duplicate]

let $u_n$ such as $u_0 > 0$ and $u_{n+1}=u_n+1/u_n$ Show that $u_n \sim \sqrt{2n}$
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0answers
41 views

Find limit of sequence defined by sum of previous terms [on hold]

I have been trying for days to find the limit of this sequence, I'm desperately hoping that someone could help find it. Any help is very much appreciated! Let a and b be natural numbers, with $1\leq ...
1
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1answer
23 views

Integral test to find constant for which series is convergent

Problem statement: Use the integral test to find the value of $b$ for which the following series is convergent $$ \sum_{r=2}^{\infty}\frac{8}{3(br+1)}-\frac{3}{2(r+1)}+\frac{1}{6(r-1)}.$$ ...
2
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3answers
60 views

How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...
0
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1answer
31 views

Is there an exception to the law of large numbers?

I was reading about the law of large numbers, and under its strong formulation it says that the sample average converges almost surely. That means that it may exist a finite subset with measure $0$ ...
0
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1answer
19 views

Prove that if $(z_n)_{n\geq 1}$ is a null sequence, then $(|z_n|^q)_{n\geq 1}$ with $\forall q\in \mathbb{Q}:q>0$ is a null sequence and vice versa.

Let $z\in \mathbb{C}, (z_n)_{n\geq 1} \subset \mathbb{C}$ be a sequence. Prove that: $(z_n)_{n\geq 1}$ null sequence $\Longleftrightarrow$ $(|z_n|^q)_{n\geq 1} \quad,\forall q\in \mathbb{Q}:q>0$ ...
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0answers
21 views

General form of a function sequence

I've problem with defining a general form of a function sequence. From the begining - last time, I was wondering, how simple function defined by other functions iside can be defined. So let we say, ...
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2answers
32 views

Find the limit of $a_n=\left(\frac{3n^4+n^2+n(-1)^n}{2n^4+i^n+n^2}\right)_n$

Find the limit of $a_n=\left(\frac{3n^4+n^2+n(-1)^n}{2n^4+i^n+n^2}\right)_n$ I assume that it tends to $1.5$ since I did some tests. For example $a_{10000}\approx 1.5$. I also know that $n^4$ ...
1
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2answers
73 views

Prove $\sum_{n=2}^{\infty}\ln\left(1+\frac{(-1)^n}{n^p}\right)$ converges if and only if $p>\frac 12$

I am trying to prove that the following sequence converges: $$\sum_{n=2}^{\infty}\ln\left(1+\frac{(-1)^n}{n^p}\right)$$ if and only if $p>\frac 12$. I've seen solutions to this exact problem here, ...
0
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0answers
72 views

Does/Can a “function” exist that can “return” “any” “number series”?

I am really a novice at Math and am interested in the existence of such function as I think it could be really useful for some experimentation in machine learning. My question is, does/can a "...
0
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4answers
31 views

If $(x_n)_{n = 1}^{\infty}$ is Cauchy, show subsequence $(x_{n_{k}})$ such that $\sum_{k = 1}^{\infty}|x_{n_{k}} - x_{n_{k+1}}| < \infty$

If $(x_n)_{n = 1}^{\infty}$ is Cauchy, show that it has a subsequence $(x_{n_{k}})$ such that $\sum_{k = 1}^{\infty}|x_{n_{k}} - x_{n_{k+1}}| < \infty$ Attempt: Since $x_n$ is Cauchy and since $\...
0
votes
3answers
41 views

How to prove that $\left(\frac{n^n}{(2n)!}\right)_{n\geq 0}$ is a null sequence. (a sequence tending to 0)

How to prove that $\left(\frac{n^n}{(2n)!}\right)_{n\geq 0}$. First of all, I realised that for $n=0$, we have an undetermined expression... Is that a possible mistake in the task? Nevermind, my ...
1
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1answer
26 views

Let $(x_n)$ be a bounded sequence and $u=\limsup x_n$. Let E be set of limits of convergent subsequences of $(x_n)$. How do I prove $u \in E$?

I've been trying to attempt this problem for a long time now. At fist I tried to show that the sequence $(u_n)$, where $u_n = \sup_{i \geq n} x_i$, is a subsequence of $(x_n)$. But this is not true ...
2
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0answers
33 views

The probability that $s$ integers selected according to the ideal soliton distribution are relatively $r$-prime

Fix integers $r,s\geq1$, not both 1. Definition. We say that integers $a_{1},\ldots,a_{s}$ are relatively $r$-prime if their greatest common divisor has no perfect $r$th power factors > 1. (When $r=...
4
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4answers
128 views

Evaluating an infinite series $\sum_{n=1}^{\infty} \frac{1}{n^{2}2^{n}}$ [duplicate]

I have been trying to find the sum of the series $$\sum_{n=1}^{\infty} \frac{1}{n^{2}2^{n}}$$ but I couldn't find any methods (such as a fourier series) that seem to get me anywhere. WolframAlpha ...
1
vote
1answer
26 views

The limit of the interval endpoints depending on $n$

For $n \in \mathbb{N_0}$ consider the sequence of intervals of the the following from: \begin{align} A_k &:= [ k, k + 1 ), \quad k \in \mathbb{N_0} \\ \frac{A_k}{2} &:= \left[ \frac{k}{2}, \...
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3answers
25 views

Need help with series index

I would like to get some help if someone explain in particular the second part of the question. I am a bit confused. I guess the first part will be $m=-2$ but don't know what will be the coefficient ...
2
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1answer
27 views

Prove geometric sum identity

Prove that ${ \sum_{n=1}^{\infty}{x e^{-nx} }} = { \frac{x}{ e^{x} - 1 } }$ for $x \in [0,1]$. I know that with geometric sums ${ \sum_{n=0}^{\infty}{x^n }} = { \frac{1}{1- x } } $but which ...
1
vote
1answer
34 views

Abel's test for uniform convergence in Fourier series

In Abel's test for uniform convergence, we wite the terms of the series as $u_n(x)=a_n f_n(x)$ and there is a condition which says that the functions $f_n(x)$ should be monotonic, in mathematical ...
2
votes
1answer
28 views

Taylor Type series

I'm stuck in the following series: $$\sum_{n=0}^{+\infty} \frac{1}{n!}\frac{d}{dx^n} \left( f(n-2x) \right) \left|_{x=0} \right.$$ where $f$ is a smooth function. At first glance it resembles a Taylor ...
0
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1answer
26 views

Why when finding a Laurent series can we factor the function

Say we are trying to find the Laurent series (at zero) of $\frac{cos(z)}{z^{3}}$ why is it we can factor out $z^{-3}$ and multiply the Taylor series expansion of $cos(z)$ at zero when we are trying to ...
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2answers
29 views

Trying to understand convergence, example problem with getting an ε

So I recently started doing convergence in my Algebra class and I am having some trouble understanding conceptually how to find the $\epsilon$-value in the definition $\left|a_n − a\right| \leq \...
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0answers
36 views

Need help clarifying one step of the proof the ratio test

A couple of days ago I submited a question asking for clarification of the proof of the ratio test connecting it with the Cauchy Hamarard Theorem. I received a lot of useful help and clarifications ...
4
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1answer
105 views

Simpler derivation of $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$ [duplicate]

I know that the equality $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ can be proved in numerous ways by using the Fourier series. However, is there a way to derive it using more fundamental ...
2
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0answers
101 views

Show that $\sum x^p$ over primes must have a non-trivial zero.

The sum $\sum x^n$ is unbounded in $|x| \le 1$. Similarly if $p$ is prime then trivially $\sum x^p$ is also unbounded in $|x| < 1$ because all primes $> 2$ are odd so the lower bound would ...
2
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0answers
50 views

$\forall$ real sequence $\left(u_n\right)_{n\in\mathbb{N}},\exists f$ analytic such that $\forall n\in\mathbb{N}, f(n)=u_n$

This might be a known result, but I haven't found anything so far. Prove or disprove: For any real sequence $\left(u_n\right)_{n\in\mathbb{N}}$, there is at least one analytic function $f$ such that ...
0
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2answers
61 views

Let $f$ be continuous prove that if $f:[a,b] \rightarrow \mathbb{R}$ then $|f(x)| \leq M$ for $M >0$

Let $f$ be continuous prove that if $f:[a,b] \rightarrow \mathbb{R}$ then $|f(x)| \leq M$ for $M >0$. Hint: suppose there exists a sequence $x_n \in [a,b]$ such that $\lim_{n \rightarrow \infty} |f(...
2
votes
1answer
38 views

Polynomial which fix set of integers with only 1 in base 10 [duplicate]

Let be $A$ the set of integers non-negative which only have 1 in their base 10 expansion (e.g. $111111$, $11$, $1$). I would like to find all polynomials $P \in \mathbb{C}[X]$ such that $P(A) \subset ...
1
vote
0answers
48 views

$\sum \frac{(-1)^n}{n(2+(-1)^n)}$ [duplicate]

Does the series $\sum \frac{(-1)^n}{n(2+(-1)^n)}$ converge or diverge. I think it diverges by looking at $a_{2n}+a_{2n+1} = \frac{1}{6n}-\frac{1}{2n+1} = \frac{1}{3}\frac{1-\frac{1}{4n}}{n+\frac{1}{2}...
9
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2answers
98 views

Calculating sum of converging series $\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n}$

I have a trouble with calculating the sum of this series: $$2+\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n}$$ I tried to split it into three separate series like this: $$2+\sum_{n=1}^{\infty}\frac{1-n}{9n^...
0
votes
2answers
58 views

When the ratio test fails.

Suppose $a_n>0$ and, for large enough $n$, $\frac{a_{n+1}}{a_n} \leq 1-\frac{c}{n}$, where $c>1$. Prove the series $\sum a_n$ converges. I tried looking at the tail of the distribution, at ...
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votes
1answer
71 views

New way of finding primes - Can it be proven wrong?

So, I believe I may have discovered a new way for finding primes, but I'm not sure if there is a definite proof; If not, then I have just created my own one. Anyway, this new method I've come up with ...
-1
votes
1answer
73 views

$x_{0}=?$ such that limit of $x_{n}$ is 2.

I have the following sequence $(x_n)_{n\geq0}; x_{n+1}=2^{\frac{x_n}{2}}; x_0$ is a real number.I need to find $x_0$ such that the limit of this sequence is $2$. Some hints?
4
votes
1answer
65 views
+50

Prouhet-Thue-Morse sequence and Arithmetic Progressions

This question is a fragment from a question posted by @Mathphile for which the link will be provided below. Let $(x_i)$ be an arithmetic progression of length $M$. Let $(t_i)$ be the $i$th element in ...
2
votes
1answer
23 views

Proving this sequence converges in $L^2(\mathbb{P})$

We have some IID sequence, $\left\{ {{X_n}} \right\}_{n = 1}^\infty $, of standard normal random variable on the probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$. Also $\left\{ {{\...
0
votes
1answer
24 views

11 x [27] = 297 & 792 ÷ 11 = [72] …why is that?

Another example of a reversed factor giving you a reversed multiple is 11 x [53] = 583 & 385 ÷ 11 = [35] Is that pattern unique to the number 11 ~ google wouldn't say :( & what is the name ...
0
votes
2answers
79 views

Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$ f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2} $$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
-1
votes
3answers
60 views

Proving that limit does not exist

How can I prove that $\lim\limits_{n \to \infty} n$ does not exist? Shall I do it by proving the sequence $\{n\}$ is not a Cauchy sequence?
0
votes
0answers
13 views

Finding the nth Padé aproximant

I am trying to find the $P_N^N(\epsilon)$ and $P_{N+1}^N(\epsilon)$Padé approximant of this function $$ x(\epsilon) = \int_0^{infinity} \frac{e^{-t}}{1+\epsilon t}dt \sim \sum_{n = 0}^{infinity} n! (-...
2
votes
3answers
65 views

$x_n$ is the sequence of all the positive roots of the equation $tan (x)=x$. Is the series $∑1/{x^2_n}$ convergent?

let ${x_n}$ be the sequence of the positive roots of the equation $tan (x)=x$ Prove/disprove: the series $∑1/{x^2_n}$ converges. I spent a lot of time trying to prove it, but I do not think I'm ...