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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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46 views

Integral into series [duplicate]

When can I put an integral into a series? what are the conditions? For example in this topic Definite integral into indefinitie series what were the conditions required?
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0answers
25 views

Conjecture on limit of a partial fraction expansion involving tan and cot

Let $q$ be an integer such that $q \equiv 1\mod{4}$ or $q \equiv 2\mod{4}$; i.e., q={... -7,-6,-3,-2,1,2,5,6,9...} I seek a proof of the following conjecture: $$(C)\quad \lim_{n \to \infty} \sum_{k=1}^...
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0answers
19 views

Convergence or divergence of a sequence with log and both radical [on hold]

Find $$ \lim_{n \to \infty} \frac{\ln n}{n^{1/n}} $$ My answer was that it is divergence. I tried this in following way taking an and equals to log n bn as second sequence equals n to the power one by ...
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2answers
33 views

Convergence of $\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$

Study the convergence of the series as $a > 0$ $$\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$$ In the text there's ...
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2answers
446 views

Real Analysis sequence limit problem

Sorry for the unclear title, the problem is too specific so I couldn't think of anything else. Here goes: If Prove that: Now, in my textbook there is a proof provided but I don't understand it. ...
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0answers
15 views

Prove that probability function $Pr(v)=\int_{G(v)} dP$ is smooth

There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters: $Pr(v)=\int_{G(v)} dP$, where $G (v)$ is the "goal" region, the probability of getting ...
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1answer
15 views

Divide 56 in four parts in AP such that the ratio of product of their extremes 1st and 4th to product of means 2nd and 3rd is 5:6

the solution is available on other sources but everyone take the four numbers as (a-3d) , (a-d) , (a+d) , (a+3d). I tried taking other numbers but failed to solve the question. I just need an ...
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2answers
41 views

Showing the convergence of the given recursively defined sequence

Let $(a_1) = 1$ and $$(a_{n+1})= \frac{4+3a_n}{3+2a_n}$$ It is required to show that the following recursively defined sequence converges. I know one way to show this converges. Define $$f(x)= \frac{4+...
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1answer
43 views

What is a binary expansion of a real number?

Related to this question I asked. I want to know what is exactly meant by a binary expansion of a number, real or natural. Can someone show me via example, how would you binary expand a real number ...
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1answer
54 views

What is $\lim_{n\to\infty}\int_{0}^\infty \exp(-x^n \arctan(\frac1x)) dx,n>1$

The following integral $\lim_{n\to\infty}\int_{0}^\infty \exp(-x^n \arctan(\frac1x)) dx$ seems converges for all integer $n>1$ according to sum computation that i runs with wolfram alpha such that ...
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1answer
18 views

$\{u_j\}$ an harmonic sequence, $\{\partial^{\alpha}, |\alpha|\le 2\}$ converges uniformly, $u = \lim_j u_j$ is harmonic in $\Omega$

Let $\Omega\subset\mathbb{R}^N$ open and $\{u_j\}$ be a sequence in $C^2(\Omega)$. If each $u_j$ is harmonic in $\Omega$ and the sequences $\{\partial^{\alpha}, |\alpha|\le 2\}$ converge uniformly ...
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4answers
44 views

Show convergence and find the limit of the sequence given by $a_1=1$ and $a_{n+1}=\frac{1}{3+a_n}$

I've been trying to solve this exam question on an exam in real analysis. Thus, only such methods may be used. The problem is as follows. Show that the sequence $a_n$ defined by $a_1=1$ and $a_{n+1}...
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0answers
67 views

Determine Convergence of $\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$

I know that $\frac{n!}{(n+1)!}$ can be reduced to $\frac{1}{n+1}$, but i'm not sure about this one. $$\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$$ In my notes, my professor reduced it to a p-...
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1answer
53 views

Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ converge?

I am shockingly terrible at determining whether or not infinite series converge or not... I'm stuck on the problem: Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ ...
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1answer
67 views

Generating function for the number of unlabeled trees on $n$ vertices

According to OEIS sequence A000055, if $(T_n)$ denotes the sequence of number of trees with $n$ unlabeled vertices, then it has the generating function $$G(x)=1+A(x)-A^2(x)/2+A(x^2)/2=\sum_{n=0}^{\...
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1answer
21 views

Uniformly convergence of a sequence defined in compact set.

Let $(f_{n})$ be a sequence of functions $[-1,1] \to \mathbb{R}$ defined by: $f_{0}(t)=0$ and $f_{n+1}(t)=f_{n}(t)+\frac{1}{2}(t^{2}-f_{n}^{2}(t))$. Show that $(f_{n})$ converges uniformly. ...
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Why does $1+2+3+4…=\frac{-1}{12}$? [duplicate]

This has always puzzled me as to why this is. I understand the argument for analytic continuation of the Reimann Zeta function, but it has always seemed like a nonsensical result that you get from ...
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32 views

Function of limit inf and limit inf of a function

Consider a Lipschitz continuous function $f : L^{\infty} \rightarrow \mathbb{R}$. Let, $\{X_{n}\}_{n=1}^{\infty}$ be a sequence on $L^{\infty}$, such that $\lim_{n \rightarrow \infty}{X_n} = X$, $X \...
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1answer
21 views

Theorem stating the connection of limits of sequences and functions.

Following is the theorem I have been given for regarding the connection between limits of sequences and functions, in order to help decide whether a limit for a function does exist or not: Theorem: ...
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1answer
29 views

Show uniform convergence of series with |x|

Let $f:\mathbb{R}\to\mathbb{R}$ be a $2\pi$ periodic function where $$ f(x)=|x|= \begin{cases} -x&x\in[-\pi,0[\\ x&x\in[0,\pi] \end{cases} $$ Show that $\sum_{n=0}^{\infty} f(2^nx)/2^n$ ...
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3answers
65 views

Sum of sequence $\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + … + \frac{2n - 1}{2^n}$ (not limit for sum) [duplicate]

Can anyone help me to find formula to sum of $n$ first elements in sequence : $$a_n = \frac{2n - 1}{2^n} $$ i.e. : $$S_n = a_1 + a_2 + ... + a_n $$ So can you explain how to find formula for $S_n$...
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1answer
56 views

Does the series for $\cos(x)/x$ converges? [duplicate]

The sequence of $$ a_x ={\cos (x)\over x} $$ does converge to zero. As a result, intuitively $$ \sum_{x=1}^\infty {\cos (x)\over x} $$ should also converge right? But I've been told that the series ...
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1answer
77 views

How to prove the two identities

Let $m,n$ be positive integers and $N=\{1,\ldots, n\}$. Try to prove the two following identites: For $m<n$, we have $$\sum_{A \subseteq N} (-1)^{\left| A\right|} \left(\sum_{j \in A} x_j\right)^m ...
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2answers
37 views

Show using the definition of convergence that the sequence 1/n does not converge to any number $L > 0$.

Let $\epsilon > 0$ $|1/n - L| < \epsilon$ for some $n\geq N$. (definition of convergence) This implies $-\epsilon\cdot n < 1 - nL < \epsilon\cdot n$ Therefore $1 - nL > -\epsilon\...
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2answers
67 views

Summing infinite series $\sum_{n=0}^\infty\frac{\cos^{2n+1}x}{2n+1}$.

How should one approach questions of this kind: What is the sum of $$\sum_{n=0}^\infty\frac{\cos^{2n+1}x}{2n+1}$$ For $x=0$ this reduces to $1+1/3+1/5+...$ which diverges by the integral test. ...
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1answer
49 views

Existence of natural number in the set $\left\{x \textrm{ } | \textrm{ } Q \leq \sin(x) < 1\right\} $, $0 < Q < 1$

Suppose $Q \in (0,1)$. Then clearly, $\exists$ $x \in \mathbb{R}$, $\ni$ $ \sin(x) = Q$. We define a set \begin{equation*} P = \left\{ x \textrm{ } | \textrm{ } Q \leq \sin(x) < 1 \right\} \end{...
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0answers
38 views

What are some series whose convergence is not known [on hold]

I found series involving division by powers of sin whose convergence is not known. But what about the following: 1) Are there series whose convergence does not depend on external unknown problems (...
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1answer
44 views

Sum of the given cosine series

How to find the sum of the following series: $$\sum_{n =-\infty}^{\infty}\cos \left(8n+\frac{2\pi}{3}\right)$$ If it was only in terms of $\pi$, I would have handled it. But I don't know how to ...
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1answer
40 views

Use Differentiability of Power Series to find the sum $F(x) = \sum_{n=0}^{\infty} \frac{(x+1)^{n+1}}{n+1}$

$$F(x) = \sum_{n=0}^{\infty} \frac{(x+1)^{n+1}}{n+1}$$ I found the following values after differentiating: $$ (x+1)^n, n(x+1)^{n-1}, n(n-1)(x+1)^{n-2} $$ It looks a lot like the Sum for $F(x)= 1/(...
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0answers
30 views

Limit of simple Fourier series $\sum_{j=1}^{\infty}\frac{1}{j^k}\cos{\frac{2\pi j}{m}}$.

Are these limits known for all $m\ge1$? $$\sum_{j=1}^{\infty}\frac{1}{j^k}\cos{\frac{2\pi j}{m}}$$ $$\sum_{j=1}^{\infty}\frac{1}{j^k}\sin{\frac{2\pi j}{m}}$$
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3answers
54 views

$\lim_{n \to \infty} a_n$, where $a_n = 2\ln(1+3n) - \ln(4+n^2)$

I'm trying to wrap my head around finding the limit as n approaches infinity of this sequence : $$a_n = 2\ln(1+3n) - \ln(4+n^2)$$ Thank you for any help!
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2answers
39 views

Tranfinite induction, 0.9…=1 [on hold]

Can transfinite induction be used to demonstrate that 0.9...=1? More generally, can it be used to prove limits of sequences?
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4answers
54 views

The series $\sum_{n=1}^\infty \frac{\sin(\pi n / p)}{n}$ converges for each $p \in \mathbb{N}$

This problem showed up on UCLA's Spring 2018 basic exam for Math Ph.D. students. The problem asks to show that for each $p \in \mathbb{N}$, the infinite series $$\sum_{n=1}^{\infty}\frac{\sin(\pi n/p)...
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2answers
23 views

Why we ONLY use ratio test and not conditional convergence to determine the interval of convergence of an alternating series?

For example, consider $$S_n=\sum_{n=1}^{\infty} \frac{(-1)^n x^n} {\sqrt{n}}$$ While determining the interval of convergence, we use the ratio test to determine the interval in which the series ...
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3answers
51 views

How to show that the sequence $(x_n) = \sin{n}$ is not monotonic?

How can I show that the sequence ($n$ is a natural number): $(x_n) = \sin{n}$ is not monotonic?
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1answer
15 views

About trigonometric test

We know that, if $b_k$ is monotone decreasing and $\lim b_k =0$, then $\sum b_k \sin kt $ is convergent for all $t\in R $. IF we change the condition from $b_1 \geq b_2 \geq ....\geq 0$ to $b_N+1 ...
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1answer
62 views

Determining whether elements map to $\aleph_0$

Consider the infinite sequences $$A=\left\{\begin{array}{lcr} 1,\\2,2,\\3,3,3,\\4,4,4,4,\\\vdots\end{array}\right\}\qquad B=\left\{\begin{array}{lcr} 1,\\2,3,\\3,4,5,\\4,5,6,7,\\\vdots\end{array}\...
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1answer
27 views

Supremum and infimum definition

I’m currently learning it from scratch and I think that I pretty got the general idea: Supremum is largest limit of a set/sequence and infimum is the smallest. I’ve encountered the following ...
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4answers
80 views

Limit As n goes to infinity of $ \sum_{n=1}^\infty e^{- \alpha n^2 }$.

I suspect the following is exactly true ( for positive $\alpha$ ) \begin{equation} \sum_{n=1}^\infty e^{- \alpha n^2 }= \frac{1}{2} \sqrt { \frac{ \pi}{ \alpha} } \end{equation} If the above is ...
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1answer
39 views

Getting a specific element of a non-recursive sequence

I have a sequence, starting with $1$. You store the current sequence as a list, then duplicate it. In this copy, you invert it, turning $1$s into $0$s, and $0$s into $1$s. Then, you join it on to the ...
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1answer
65 views

What is the series expansion of the $n$-th derivative of this : $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\text{erf}(x)}dx$

$\newcommand{\erf}{\operatorname{erf}}$ The computation of $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\erf(x)}dx$ with wolfram alpha we have for $n=1, n=2, ..n=4$ interesting expansion which seems present ...
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2answers
70 views

Sum of the series $\frac{1}{2.4.6}+\frac{2}{3.5.7}+\frac{3}{4.6.8}+…+\frac{n}{(n+1).(n+3).(n+5)}$.

Sum to n terms and also to infinity of the following series: $$\frac{1}{2.4.6}+\frac{2}{3.5.7}+\frac{3}{4.6.8}+.....+\frac{n}{(n+1).(n+3).(n+5)}$$ the solution provided by the book is $$S_n=\frac{...
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0answers
31 views

How to show that $\exists N \in \mathbb{N}$ s.t $|a_n|^{1/n} < \frac{1}{R} <\frac{1}{r}$, where $\limsup |a_n|^{1/n} = \frac{1}{R} $

In the book of Functions of One Complex Variables by Conway, at page 31, it is claimed that However, I cannot understand the existence of such an $N$ that makes $|a_n|^{1/n} < 1/r$. I mean as far ...
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2answers
45 views

How can we prove that $z^{n+1} \to 0_{\mathbb{C}}$ as $n\to \infty$?

In the book of Function of One Complex Variable by Conway, at page 31, it is given that However, normally, if $z$ was a real number, we could argue that $z^{n+1}$ goes to zero as $n \to \infty$ if $|...
1
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2answers
55 views

Why the sequence $\frac{2+a_n}{\sqrt{2+a_n^2}}$ is bounded

Given $\left\{ a_{n}\::\:n=1,\:2,\:3,\:\cdots\right\} $ is an infinite sequence in $\mathbb{R}$, and every term is positive. How to prove that the set \begin{equation} \left\{ \frac{2+a_{n}}{\sqrt{2+...
0
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1answer
27 views

Prove that the function can be continued into a larger domain

Prove that the function $f(z)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{z^n}{n}$ can be continued into a larger domain by means of the series $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
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4answers
57 views

Is $\sum_{n=1}^{\infty}\frac{\sin a_{n}}{\sqrt{n}+na_{n}}$ convergent?

Suppose $a_{n}>0$ and $\sum_{n=1}^{\infty}a_{n}$ is convergent. $\sum_{n=1}^{\infty}\frac{\sin a_{n}}{\sqrt{n}+na_{n}}$ convergent ? Since $\sin a_{n}$ is bounded by one, and $\sqrt{n}\...
20
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2answers
398 views

Does recursively replacing $\frac1n$ by $\frac1n(\frac12+\dots+\frac1{n+1})$ really converge to $\frac1e$?

I was thinking of a problem and have an answer through computer programming, but am unable to prove it mathematically. Start with the following: $$\frac{1}{2}\bigg(\frac{1}{2} + \frac{1}{3}\bigg)\...
5
votes
2answers
89 views

Why $\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2}$ converges to $\sum_{k=1}^\infty e^{-2\pi^2k^2}$?

While I was considering random walk on $S^1$, I needed to compute the following. $$\lim_{m\rightarrow\infty}\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2}$$ I guessed it should ...
5
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2answers
210 views

A nice expression for $\int_0^{\pi/2} \left[\frac{1}{x \sin(x)}-\frac{1}{x^2}\right] \mathrm{d} x$

Motivated by the easier integral $$ \int \limits_0^\infty \left[\frac{1}{x^2} - \frac{1}{x \sinh(x)}\right] \mathrm{d} x = \ln(2) \, ,$$ I have been trying to compute $$ I \equiv \int \limits_0^{\...