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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26 views

Limit with sin function

Here(Task with combination of spectrums of matrices) I continued to post tasks from an old notebook. So there is the next one: Let $|a|\le\frac{\pi}{2}, x\in\mathbb{R}$. Then $x_0=a\sin{x}, \hspace{0....
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2answers
30 views

A series convergence

What is the closed-form answer for this series(if there is): $$f(z)=\sum_{n=1}^{\infty}\dfrac{z^n}{n^2(n+1)}$$ assuming $z$ is a complex number and $|z|<1$. I have looked for some approximating ...
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1answer
24 views

Given a single die, what is the probability it takes an even number of rolls to get a 4? [duplicate]

So I've decomposed the probability as meaning we failed $2k-1$ times and succeeded on the $2k^{th}$ roll. I set up a series for this: $\sum_{k=1}^{\infty}\frac{1}{6}\cdot\left(\frac{5}{6}\right)^{2k-...
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0answers
14 views

Order of convergence of a product of two convergent sequences

Let $a_n$ be a sequence that converges to $A$ with order of $n^\alpha$, that is $a_n = A + \mathcal{O}(n^\alpha)$ and $b_n$ is another sequence that converges to B with order of $n^\beta$; i.e. $b_n = ...
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1answer
25 views

Show a series in $\ell^{\infty}$ converges to a point in $c_0$

Let $x =(x_1, x_2, ...) \in \ell^{\infty}$ Suppose $\displaystyle\sum_{n=1}^{\infty}x_n e_n$ converges to $x$ with respect to $\| \cdot\|_{\infty}$. Show that $\displaystyle\lim_{n \rightarrow \infty}...
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1answer
26 views

Finding common ratio of a geometric series from a cubic

The question is asking to find the sum of a geometric series up to term n, where the first value of r is 1. The expression is $$ \sum_{r=1}^n \ 4r^3-3r^2+r$$. I am aware of the general formula for the ...
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1answer
13 views

Epsilon-N method: proving itertative sequence

How do you prove an iterative sequence like $$x_{n+1} = ax_n + b$$ $$ 0< |a|< 1 $$ by using epsilon-N method? I tried create a closed form of the sequence $$a^{n+1}x_0 + b\sum_{i=0}^n a^i$$. ...
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0answers
12 views

what is the limit of this sequence?. detailed steps please [duplicate]

enter image description here Most probable limit seems infinity. I tried squeeze theorem and I was able to come to an understanding of infinity. But I want to know more detailed steps, to have more ...
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4answers
90 views

Prove this formula $\dfrac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=-\infty}^{+\infty}r^{n}\cos\left(nx\right)$

I am trying to use prove, by just simple algebraic manipulation, to prove the equality of this formula. $$\dfrac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$$ I have ...
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1answer
30 views

Compute the difference of three series (motivated by an integral formula)

Let \begin{align} L&=2^{3 / 2} \pi^{4} \frac{1}{4} \sum_{n, m, k=1}^{\infty} n^{2} m a_{n} a_{m} b_{k}\left(\delta_{m, n+k}+\delta_{n, m+k}-\delta_{k, n+m}\right) \\[5pt] R_{1}&=2^{3 / 2} \pi^{...
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1answer
37 views

Prove this sequence converge to 0.5?

prove that, as k -> infinity $$\sqrt k \cdot \sqrt {k + 1} - k \leqslant 0.5$$ I tried following $$\mathop {\lim }\limits_{k \to \infty } \left( {\sqrt k \cdot \sqrt {k + 1} - k} \right) = 0.5$...
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0answers
19 views

rearrangement of series in $\ell^p$ space

Let $x \in \ell^p$, where $1 \leq p < \infty$. Notice that we can write $x= \sum_{n=1}^{\infty} x_n e_n$, where $e_n=(0, \cdots, 1,0, \cdots )$ i.e. $1$ is at the nth term. I want to show that that ...
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1answer
74 views

How to express this sequence $a_n$ in math notation?

How do I express this sequence $a_n$ in math notation where each pair $(x,y)$ is coprime? Examples $a_2=\{(1,1)\}$ $a_3=\{(1, 2), (2, 1)\}$ $a_5=\{(1, 4), (2, 3), (3, 2), (4, 1)\}$ $a_7=\{(1, 6), (2, ...
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1answer
40 views

Why do we need that $f$ is continuous at $L$ to have $\lim \limits_{n \to \infty} f(a_{n}) = f(L)$ when $\lim \limits_{n \to \infty} a_{n} = L$?

If $$\lim \limits_{n \to \infty} a_{n} = L $$ and the function $f$ is continuous, then $$\lim \limits_{n \to \infty} f(a_{n}) = f(L)$$ I do not understand why do we have to indicate that $f$ is ...
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2answers
27 views

Would this proof by induction for the sequence of the product of odd numbers be correct? Is there something I can do better with it?

I'm new to proofs by induction, and I was challenged by my teacher to learn this when I came up with a form to represent this sequence. Would this be correct? $$1\cdot3\cdot\ldots\cdot(2n-1)=\frac{(2n)...
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0answers
27 views

Can we find generating function for $\sum_{k=0}^\infty \sum_{i=0}^n(-1)^i\binom{n}{i}\binom{rk-mi-1}{n-1}$?

For $n,m,r\in\mathbb{N}$, let $A_n(m,r)$ be number of $n$-tuples, $(x_1,x_2,...,x_n)\in\mathbb{Z}^n$ such that $1\leq x_1,x_2,...,x_n\leq m$ and $r$ divides $x_1+x_2+\cdots+x_n$ I can find that $$ ...
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1answer
25 views

What can we conclude about the convergence and uniform convergence of an infinite series, if it's radius of convergence is infinite?

Question: What can we conclude about the convergence and uniform convergence of an infinite series, if it's radius of convergence is infinite ? For example: Let us consider an infinite series $~\sum_{...
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0answers
19 views

Simplifying a geometric series with three distinct terms

The question is really quite simple - I would like to simplify the following series: $\sum_{z=1}^{\infty}\sum_{y=1}^{\infty}\left(\frac{1}{2}\right)^y(1-e^{-y})^2e^{-y(x+z-2)}$ My approach: It is ...
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0answers
26 views

A curious double integral on the unit square related to $\psi(z)$

I came along the following double integral in a statistics problem: For $z>0$ $$ I(z,s)=\int_0^1\int_0^1\left(1-\frac{(1-x)(1-y)}{(1-(1-z)x)(1-(1-z)y}\right)^{s-2}\,\mathrm dx\mathrm dy. $$ All I ...
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0answers
16 views

Interchange Lebesgue-Integral and infinite Sum with $f_n=\chi_{[n,n+1)}-\chi_{[n+1,n+2)}$

Let $f_n \colon \mathbb{R} \to \mathbb{R}$, $f_n = \chi_{[n,n+1)}-\chi_{[n+1,n+2)}$ where $\chi$ is the indicator function. Have I calculated the following statements correctly? $\lambda$ is the ...
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0answers
16 views

Find an expression for the total distance using sequence

For n = 1, d(1) = l n = 2, d(2) = l + 4l n = 3, d(3) = l + 4l + 8l d(n) = l + 4l + 8l +.....+ (n-1)(4l) how to find the general equation
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0answers
33 views

Can I get a geometric intuition to what a vercongent sequence is

I saw in a textbook, the definition of a vercongent sequence, but I was not really giving a way to understand a from a geometric point of view.
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1answer
45 views

How to calculate the limit of zeta function

Suppose $f(x)>0$,$f''(x)\leqslant0$,and $\lim\limits_{x\to+\infty}f(x)=+\infty$ on$[0,+\infty)$.prove that $$\lim\limits_{s\to0^+}\sum\limits_{n=0}^{\infty}\dfrac{(-1)^n}{f^s(n)}=\frac{1}{2}.$$ I ...
2
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2answers
26 views

Bounded Sequence help

A sequence {$S_n$} is defined as follows: $S_1$ = 1/(3+1), $a_2$ = 1/(3+1) + 1/($3^2$+2), $S_n$ = $\sum_{n=1}^\infty$ 1/($3^n$+n) Evidently this sequence is increasing as for every $a_n$ is less than $...
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0answers
24 views

kth term of summation of summation problem [closed]

so the question is $$5+\Sigma_5+[\Sigma_5+\Sigma_4+\Sigma_3+\Sigma_2+1]+[\{\Sigma_5+\Sigma_4+\Sigma_3+\Sigma_2+1\}+\{\Sigma_4+\Sigma_3+\Sigma_2+1\}+ \{\Sigma_3+\Sigma_2+1\}+\{\Sigma_2+1\}+\{1\}]\dots$$...
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1answer
34 views

Showing that $\sum_{r=1}^{N} \exp\left(\frac{2\pi irs}{N}\right)=0$ when $0<s<N$

I am trying to show that $$\sum_{r=1}^{N} \exp\left(\frac{2\pi irs}{N}\right)=0$$ when $0<s<N$. This result is stated in a paper by A. Turing (see page $39$). I can see how it is a geometric ...
2
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0answers
34 views

Showing a biconditional statement about function lim sups in $\Bbb R^n$, and codifying the intuition into a proof

$ \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\BB}{\mathcal{B}} \newcommand{\ve}{\varepsilon} \newcommand{\para}[1]{\left( #1 \right)} \newcommand{\set}[1]{\left\{ #1 \right\} }...
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4answers
36 views

Show that the sequence $x_n$ converges.

$$x_n = 1 + \frac{2}{4} + \frac{3}{4^2} + \dotsb + \frac{n}{4^{n-1}}$$ So I need to see if the sequence is monotone and bounded. By doing $x_{n+1} - x_n$ I get $\frac{n+1}{4^n} > 0$ so its ...
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0answers
29 views

The sequences $a_n = \sqrt[n]{4^nn}$ is converge or diverge? Find the limit of each convergent sequence. [duplicate]

The sequences $a_n = \sqrt[n]{4^nn}$ is converge or diverge? I don't know how to determine the sequences is converge or diverge and find the limit of each convergent sequence. Can someone help me to ...
2
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3answers
64 views

Proof if $f$ is continuous then the preimage is closed.

I want to show that if $f: \mathbb{R}\to\mathbb{R}$ is continuous then $f^{-1}(l) = \{x\in\mathbb{R}|f(x) = l\}$ is closed. I am not sure of my proof as I feel like I am missing a step: I decided to ...
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2answers
53 views

Help understanding proof of “$\sum |a_n|$ converges $\Rightarrow$ $\sum a_n$ converges”

I'm self-studying Real Analysis. I can't understand a proof of this commonly known theorem: Theorem. Every absolutely convergent series is convergent. Proof. If $\sum |a_n|$ converges, given an ...
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1answer
69 views

How did Euler obtain these two formulae? Are they correct?

I am reading this book on trigonometric series, "Тригонометрические ряды от Эйлера до Лебега" (Trigonometric series from Euler to Lebesgue) , it is in Russian, and my Russian is abysmal. But ...
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0answers
31 views

convergence test for this sin series [closed]

$$\sum_{n=1}^∞\frac{2n^3+7}{n^4 sin^2⁡(n)}$$ How to test convergence using comparision test?
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0answers
19 views

Comparison test for a series of sine and cos [closed]

$$\sum_{n=0}^∞ \frac{2^n sin^2⁡(5n)}{4^n+cos^2⁡(n)}$$ HOw to test convergence by using comparision test
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1answer
44 views

Which convergence test should I use to test this functional series $\sum_{n=0}^\infty \cos(nx)$

The following series $$\sum_{n=1}^\infty (-1)^ncos(nx)=\cos(x)-\cos(2x)+\cos(3x)...$$ is obviously divergent when $x=0$. At that point, we have the Grandi series $1-1+1-1+1...$ when $x=\pi/2$, the ...
1
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1answer
37 views

Counting chessboard rectangles

I am looking into the puzzle count the number of rectangles in a regular $8*8$ chessboard. For a 1 by 1 chessboard there are 0 rectangles For a 2 by 2 chessboard there are 4 rectangles (2 by 1) For a ...
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0answers
21 views

Let $g_n=\sum_{i=1}^n a^{n-i}x_i$. Prove that $g_n\to 0$.

Let $0<a<1$ , $(x_n)\to 0$ , $x_n >0 , \forall n\in \mathbb{N}.$ Let $g_n=\sum_{i=1}^n a^{n-i}x_i$. Prove that $g_n\to 0$. Here is the sketch of my proof: We want to show that $|g_n|<\...
2
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1answer
30 views

Question about continuity of linear transformations

Assume $ W,V $ are two normed vector spaces over $ \mathbb{C} $ and let $ T:V\to W $ be a linear transformation. Prove that $ T $ is continuous if and only if for any sequence of vectors $ (v_n)_n $ ...
2
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3answers
55 views

Knowing that $f(n)=n$, $g(n)=n\sqrt3$. Find $h(n)$ that contains all elements of $f(n)$ and $g(n)$ in ascending order.

Knowing that: $n$ is an integer going from zero to infinity, $f(n)=n$, $g(n)=n\sqrt3$. I need a formula for $h(n)$ that can generate the series: $0, 1, \sqrt3, 2, 3, 2\sqrt3, 4, 5, 3\sqrt3, 6, 4\...
1
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1answer
22 views

How to consider “and” between quantifiers?

Let $\{x_n\}$ be a sequence in $\mathbb{R}$ and $a \in \mathbb{R}$. How would I negate $\exists \varepsilon>0$ and $\exists N \in\mathbb{N}$ such that $\forall n\ge N$, $|x_n - a| \ge \varepsilon$ ...
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0answers
14 views

Does $\sum_{k=0}^\infty \frac{2k!}{(k!)^2}\cdot\frac{1}{4^k}$ converge or diverge? [duplicate]

Does $\sum_{k=0}^\infty \frac{(2k)!}{(k!)^2}\cdot\frac{1}{4^k}$ converge or diverge. I tried to test this with the comparison test but I am stuck because I don't know if $(k!)^2 <(2k)!$ holds for ...
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0answers
34 views

calculate $\sum_{n=1}^{\infty} \frac{1}{(a+n)(b+n)}$ [duplicate]

How to calculate $$\sum_{n=1}^{\infty} \frac{1}{(a+n)(b+n)}$$ where $0 <a,b< 1$?
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3answers
55 views

$\lim_{n \to \infty} \left(1-\frac xn+\frac{x}{n^2}\right)^n = e^{-x}$

How can we mathematically precisely argue that $$\lim_{n \to \infty} \left(1-\frac xn+\frac{x}{n^2}\right)^n = e^{-x}$$ holds? So how can we bring $$1-\frac xn+\frac{x}{n^2} = 1- \frac{(n+1)x}{n^2} \...
0
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0answers
22 views

Radius of convergence of sum of two complex series

$$\sum\limits_{n=0}^{\infty}\frac{1}{n^n(z-2+i)^n}+\sum\limits_{n=0}^{\infty}(1+in)(z-2+i)^n=I_1+I_2$$ My first idea was to examine radius of convergence for each series, but I don't know if it is a ...
-4
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0answers
31 views

True or False? If a_{n}> 0 e b_{n}> 0.. [closed]

True or False? If $a_{n}>0$ and $b_{n}>0$, and $\lim_{n\to\infty}(a_n*b_n)>\infty$, then $\lim a_n>\infty$ or $\lim b_n>\infty$
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3answers
40 views

Understanding how different groupings of terms in an Infinite series can lead to different answers.

One of the most, conceptually speaking, for me to understand is the topic of infinite series. I have always had a hard time proving that an infinite series diverges or even finding a solution for the ...
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1answer
31 views

How to test whether a series converges or diverges

I've tried every convergence criterion I know. (Dalamber's, Cauchy's and Raabe's tests) But in all cases, because of $-1^n$, I got indeterminate form. $$\sum_{n=1}^\infty (-1)^n \sin(\sqrt{n^2-1} - 1)$...
1
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0answers
22 views

Exercise on radius of convergence of power series

I have solved the following exercise and I would like to know if I have made any mistakes: Let $\sum a_n x^n$ be a power series with $a_n\neq 0$ and assume $L=|\frac{a_{n+1}}{a_n}|$ exists. (a) Show ...
1
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2answers
47 views

I have to show that the sum of this double series is $\frac{1}{2}$ [duplicate]

i have to solve this double series. i tried it, but i am not sure, that it is enough. $$\sum_{i=1}^{\infty} \sum_{k=1}^{\infty} \left(\left(\frac{1}{k+1} \cdot \left(\frac{k}{k+1}\right)^{i}\right) - \...
3
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1answer
65 views

Relation between Rogers Ramanujan continued fraction and $j$-invariant

While going through this answer I found an interesting but slightly complicated relation between Rogers-Ramanujan continued fraction and the j-invariant. I would like to know an elementary proof of ...

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