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Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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Generalized of Miklos Schweitzer 1980 P1

question; For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n=1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there ...
zhihu's user avatar
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1 answer
36 views

Problem with the convergence region of the series

Can the region of convergence of a functional series defined on the real axis consist of a half-interval and a segment? Yes maybe! I understood this intuitively, but I couldn’t show it clearly. I ...
Dmitry's user avatar
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2 votes
1 answer
51 views

$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0$?

The fractional part is defined of a positive real number $x$ is defined by $\{x\}:=x-\lfloor x\rfloor$. Is it true that $$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0 \ ?$$ If so, ...
Nathan Portland's user avatar
0 votes
0 answers
25 views

calculate$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$

I am wondering how to calculate the following expression: $$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$$ I have tried to use the Jacobi-Anger Expansion, also the equation below: $$\sum_{n=-\...
madao's user avatar
  • 1
0 votes
1 answer
60 views

Is the following sequence bounded?

I want to prove the complexity of an algorithm I designed. To do this I need the following to be true: Let $f(n)$ be a function in both $\Omega(n^{-5/2})$ and $\mathcal{O}(n^{-3/2})$ and let $\alpha$ ...
Wannes De Maeyer's user avatar
3 votes
1 answer
93 views

Calculate $\sum\limits_{n=0}^{\infty}{\int\limits_{\frac{1}{2}}^{\infty}(1-e^{-t})^{n}e^{-t^2}dt}$

$$ \mbox{Calculate}\quad \sum_{n = 0}^{\infty}\int_{1/2}^{\infty} \left(1 - {\rm e}^{-t}\right)^{n}{\rm e}^{-t^{2}}{\rm d}t $$ Basically I don't know where to start. I was thinking of using ...
Ranko's user avatar
  • 189
1 vote
0 answers
34 views

radius of a convergent sequence

I have already finde r for den first one, is there anyone who can help me with second part? (a) Show that the power series $ \sum_{n=0}^\infty \frac{n^2}{n!} x^n $ has a radius of convergence ...
user1321504's user avatar
0 votes
0 answers
41 views

Cool identities/properties involving the Alternating Harmonic Numbers

Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
Kamal Saleh's user avatar
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0 votes
1 answer
58 views

Application of Bertrand's postulate [duplicate]

We can use Bertrand's Conjecture ( that for any integer $n \not= 0$, there exists at least one prime number $𝑝$ with $n < p \leq 2n$ ) to demonstrate the ...
Alexis J's user avatar
1 vote
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2 concise tables of “usual” series (mostly trigonometrics) and of "usual" L-series (Zeta, Eta, Beta...)

CONTEXT Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ ...
olivierlambert's user avatar
1 vote
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Can I substitute $\cos^{(k)}(0)+i\sin^{(k)}(0)$ for $i^k$ in a series with index k?

I was trying to come up with a derivation of Euler's Formula inspired by a section in Thomas' Calculus, and I am using the following equation in this derivation $$\sum_{k=0}^{\infty}\left(\frac{\cos^{(...
Matti Bock Guldager's user avatar
-3 votes
0 answers
24 views

$ I\left(\bigcup^{\infty}_{i} A_i\right) = I(A_1) + I(A_2)(1 - I(A_1)) + I(A_3)(1 - I(A_1))(1 - I(A_2)) + \ldots$ [closed]

$ I\left(\bigcup^{\infty}_{i} A_i\right) = I(A_1) + I(A_2)(1 - I(A_1)) + I(A_3)(1 - I(A_1))(1 - I(A_2)) + \ldots$ I want to prove this equality. I do not know how to approach it. I want to prove this ...
mike's user avatar
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0 answers
23 views

If $|S|=n$, series related to the number of ordered pairs $(A,B)$ such that $A\subseteq B\subseteq S$. [duplicate]

While solving if $|S|=n$, find the number of ordered pairs $(A,B)$ such that $A\subseteq B\subseteq S$, I derived this interesting series $$\sum_{i=0}^{n}\left(\frac{n!}{i!\left(n-i\right)!}\sum_{j=0}^...
Aryan Kumar's user avatar
5 votes
0 answers
120 views

Boundedness of the sequence $\left\{\sum_{k=1}^n \frac{1}{k}\sin\frac{n}{k}\right\}$.

I want to study the boundedness of the sequence $\left\{\sum_{k=1}^n \frac{1}{k}\sin\frac{n}{k}\right\}$. In one of my old post $\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{k}\sin\frac{n}{k}$, we got that: ...
Riemann's user avatar
  • 7,624
-2 votes
0 answers
35 views

Real analysis supremum of a set [closed]

Define supremum of a set $S\subset R$ and show that the $\sup S =\frac{1}{2}$ where $S=\{\frac{n}{2n+1} \mid n\in N\}$.
Hamzzy omar's user avatar
2 votes
2 answers
90 views

how do I find all $k \in \mathbb{R} $ such that $\lim_{n \to \infty} a_n = +\infty$ for a given sequence without using approximations

I have a sequence $(a_n)_{n \in \mathbb{N}}$ of positive terms defined by the recurrence relation $ \frac{a_{n+1}}{a_n} = \left(\frac{2n}{2n + k + 4}\right)^{2n}, $ where $k \in \mathbb{R}$ and $k \...
Renato German Chavez Chicoma's user avatar
2 votes
1 answer
68 views

Evaluating the integral in an integral representation of the sum that is the Fourier series for $e^{ax}$

I aim to evaluate the following sum through an integral representation of it: \begin{align*} & \frac{\sinh a \pi}{\pi} \sum_{n=-\infty}^\infty \dfrac{(-1)^n}{a - i n} e^{i nx} \nonumber \\ & = ...
Dave77's user avatar
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0 answers
11 views

Is $C_i$ monotonically decreasing, and does it possess an analytical solution?

Let $$\alpha_i = \frac{b+i}{b+i^a}, 0<a\leq 1$$ and $$\alpha_t^i = \alpha_i \prod_{j={i+1}}^{t}(1-\alpha_j)$$ I aimed to derive the analytical expression for $$C_i = \sum_{t=i}^{\infty}\alpha_t^i$$ ...
Zhou_Key_Error's user avatar
0 votes
1 answer
19 views

Aggregates and Psuedorandomness

Psuedorandomness in compsci usually takes the form of Math.random — in binary it is theoretically possible to get a random-looking number from the method: Math.random(MaxInteger) is expected to return ...
tylerbakeman's user avatar
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0 answers
118 views

Convergence of the sequence of the orthocenters, incenters and centroids of a triangle.

Now asked on MO here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is ...
pie's user avatar
  • 5,625
4 votes
1 answer
94 views

Alternating series test to prove the convergence of the series

I want to apply the alternating series test to prove the convergence of the series $\sum_{n\geq 1} (-1)^n u_n$ where $u_1=1$ and $\forall n\in \mathbb{N},\quad u_{n+1}=\frac{\cos(u_n)}{n^\alpha}$ ...
Noname's user avatar
  • 559
-2 votes
0 answers
26 views

Let $\prod_{i=1}^np_i^{a_i\cos^2(a_i)} = 2019$, then [closed]

Let $$\prod_{i=1}^np_i^{a_i\cos^2(a_i)} = 2019$$ where $p_i$ denotes the $i$-th prime number ($p_1=2 , p_2=3, p_3=5$, etc. up to $p_n=2017$), and $a_i\in\Bbb R$ How can we prove the four following ...
Hitesh's user avatar
  • 67
4 votes
2 answers
65 views

Limits of arbitrary polynomials divided by constants greater than 1 raised to the power of n

Problem statement: Prove that for any $a > 1$ and any polynomial $p(x)$, we have: \begin{align*} \lim\limits_{n \to \infty} \dfrac{p(n)}{a^n} =0 \end{align*} Solution: We need to use squeeze ...
Ragemprand Hrekt's user avatar
-2 votes
0 answers
12 views

sequence of independent random variables with uniformly bounded variances [closed]

Every sequence {X,,} of independent random variables with uniformly bounded variances obeys : (a) Borel-Cantelli lemma. (b) Cauchy’s criterion. (c) WLLN, (d) SLLN
bymi's user avatar
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1 vote
0 answers
26 views

About the converse of below theorem

My book states that if a sequence $a_n$ converges to $l$, then every subsequence also converges to $l$. Now it states at the end that its converse is not true. but as we know that if every ...
Fayaz' Rasool's user avatar
0 votes
1 answer
44 views

Investigate the convergence of $\sum_{n=1}^\infty kv_n$ and $\sum_{n=1}^\infty (u_n+v_n)$ when $\sum_{n=1}^\infty v_n$ does not converge [closed]

We already know that if $\sum_{n=1}^\infty u_n $ and $\sum_{n=1}^\infty v_n $ are both convergent, then $\sum_{n=1}^\infty ku_n $,$\sum_{n=1}^\infty kv_n $ ($k$ is constant) and $\sum_{n=1}^\infty (...
Lumiat's user avatar
  • 1
1 vote
1 answer
41 views

An example application where convergence in probability is useful or valuable

I have been learning different notions of convergence for sequences of random variables. I know the almost sure convergence is the strongest. The next best thing is convergence in probability. Can ...
curiosity's user avatar
  • 147
-4 votes
1 answer
59 views

Need help with generalized binomial formula (a+b)^n where neQ [closed]

Something is really bugging me about the binomial expansion with rational exponents. Maybe someone can explain this to the non-math person that I am. Say we are interested in the first 4 terms in the ...
Christian Scheibner's user avatar
2 votes
0 answers
19 views

Definition and Use of the Schett Polynomial in the Jacobi Taylor Series

I am having a tough time understanding the definition and use of the Schett polynomial introduced in the paper here. I have two questions related to this polynomial. My first question concerns its ...
Kyler Rusin's user avatar
1 vote
0 answers
69 views

Showing sequence converges knowing its bounded and $a_{n+1} - a_n \geq-\frac1{2^n}$ for all $n$ [closed]

Let $(a_n)$ be bounded sequence, s.t for all $n \in \mathbb{N}$, $a_{n+1}\geq a_{n} - \dfrac{1}{2^n}$, I'm stuck on showing $(a_n)$ converges.
csmathstudent8's user avatar
1 vote
0 answers
17 views

Multiplication of convergent sequence and multiplication of series

It is a basic rule that $((\lim_{n\to \infty}a_{n}=A\in\mathbb{R})\wedge (lim_{n\to \infty }b_{n}=B\in\mathbb{R})\Rightarrow lim_{n\to\infty}(a_{n}b_{n})=AB)$ And according to the definition of ...
cycle motor's user avatar
3 votes
0 answers
48 views

What is wrong with this proof of the Bolzano-Weiserstrass theorem?

I tried proving the Bolzano-Weierstrass theorem before looking at the solution in my math textbook. This is what I tried: Take a bounded sequence $x_n$. Every sequence has a monotone subsequence ...
user3612's user avatar
0 votes
1 answer
64 views

Formula for Root Series

Given the series: $S_n=\sqrt{c}+\sqrt{c-1}+\sqrt{c-2}+\sqrt{c-3}+...+\sqrt{c-n}$ in which $c$ can be any value but independent from n. Is there a formula for these kinds of series? Excluding the ...
Anthony's user avatar
0 votes
0 answers
23 views

Two series converges pointwise to same limit, and one converges uniformly

We have $\sum{f_n}$ and $\sum{g_n}$. $f_n,g_n$:[0,1]->$\mathbb{R}$ and all f are non-negative and all g are continuous. Both series converges pointwise to the same limit. How to prove that if $\sum ...
AveriX's user avatar
  • 19
2 votes
1 answer
39 views

Recursion with odd options [duplicate]

Consider a function $f(n)$ defined from $\mathbb{N}\to \mathbb{Q}$ such that $f(n+1)= f(n) + \frac{1}{f(n)^2}$. Given that $f(1)=1$, which of the following options are correct? $f(9000)>30$ $f(...
A shubh's user avatar
  • 141
-5 votes
0 answers
47 views

Trouble in understanding the series $\sum_{n = 1}^\infty \left[\frac{4}{n(n + 1)} - \frac{1}{3^n} \right]$ [closed]

Find the sum of the series $$\sum_{n = 1}^\infty \left[\dfrac{4}{n(n + 1)} - \dfrac{1}{3^n} \right]. $$ I can't seem to end on a finite sum for this series when you split it into twos because the $-1/...
Kermitheweeb's user avatar
1 vote
1 answer
37 views

Compute $\int_{\phi_1}^{\phi_2} \left( \sum_{n=0}^N a_n \cos(n \phi) \right)^2 \,\text{d}\phi$ analytically?

For a computational model I need to evaluate the following integral many times: $$ I(\phi_1, \phi_2) = \int_{\phi_1}^{\phi_2} \left( \sum_{n=0}^N a_n \cos(n \phi) \right)^2 \,\text{d}\phi, $$ where I ...
NickFP's user avatar
  • 179
0 votes
0 answers
40 views

Find a bounded real sequence, whose range has exactly one accumulation point, but which is not convergent [duplicate]

Example of bounded sequence on $ \mathbb{R}$ s.t. the set $\{X_n : n \in \mathbb{N}\}$ has exactly one accumulation point but $X_n$ is not convergent My thoughts: Since Xn is bounded it has ...
GaloisRocks's user avatar
2 votes
2 answers
105 views

How to find the summation of this following series.

Q) How to find the summation of the series in a shortcut method: $\frac{1}{\prod_{i=1}^{7}i}+\frac{1}{\prod_{i=2}^{8}i}+\frac{1}{\prod_{i=3}^{9}i}+...+\frac{1}{\prod_{i=22}^{28}i}$ I know how to find ...
Dropper's user avatar
  • 119
-1 votes
0 answers
93 views

Arc to Chord ratio as an infinite sum: a relation between Madhava-Leibniz and Newton formulas for Pi

CONTEXT From Newton $$\frac{{\pi}}{2\sqrt2}=\frac{1}{1}+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{13}-\frac{1}{15}+\cdots$$ From Madhava-Leibniz $$\frac{\pi}{4}=\frac{1}{1}-...
olivierlambert's user avatar
2 votes
0 answers
34 views

Interesting Infinite Product Involving Subfactorials (Dearrangements)

I'm trying to compute: \begin{gather*} P =\lim_{n \to \infty} \prod_{i = 2}^n \left ( 1 - \frac{(-1)^{i-1}}{S_i} \right ) \end{gather*} Where $S_i$ is given by: \begin{gather*} i!\sum_{j = 1}^{i-1} \...
NEON's user avatar
  • 196
-2 votes
0 answers
20 views

Partial sum expresion for $ S = \sum_{n=1}^{x} k^{\frac{1}{n}} $ [closed]

I was playing around with some series when I encountered this problem. I want to find a closed form partial sum for this series: $$S = \sum_{n=1}^{x} k^{\frac{1}{n}},$$ where $k$ is any constant. I ...
 Rutajit45 a maths guy's user avatar
9 votes
1 answer
296 views

Prove that this sequence is eventually one

Let $ \ \mathbb{N} = \{ 0,1,2,3,4,...\} \, $, $ \ O = \{ n \in \mathbb{N} : n \text{ is odd} \} \ $ and $ \ T: O \to O \ $ be such that, for all $ \ n \in \mathbb{N} \, $, \begin{align*} T(8n+1) & ...
Gustavo's user avatar
  • 2,106
0 votes
3 answers
100 views

Doubts about the divergence of $\sum_n\left(\frac{2n}{2n+1}\right)^n \sqrt{\frac{1}{2n+1}}$

Let $$y_n:=\left(\frac{2n}{2n+1}\right)^n \sqrt{\frac{1}{2n+1}}.$$ The series $$ \sum_{n=1}^{\infty}y_n$$ diverges by using the limit comparison test with $\sqrt{\frac{1}{2n}}$, but I seem to prove ...
Morten's user avatar
  • 629
2 votes
2 answers
93 views

Prove that $\exp \left(\dfrac{-2 \sum_{n=0}^{K-1} \frac{2^n}{n!}}{ \sum_{n=0}^K \frac{2^n}{n!}}\right) \sum_{n=0}^{K-1} \dfrac{2^n}{n!} -1 \geq 0$

I want to show the following inequality: $$\exp \left(\dfrac{-2 \displaystyle \sum_{n=0}^{K-1} \frac{2^n}{n!}}{\displaystyle \sum_{n=0}^K \frac{2^n}{n!}}\right) \displaystyle \sum_{n=0}^{K-1} \dfrac{2^...
Jane's user avatar
  • 23
0 votes
1 answer
71 views

Show that the series $A_n$ converges

Let $\operatorname{f}_{n}\left(x\right) = x^{n}\sqrt{1 - x}$ and $A_{n}$ be the area bounded by the $x$-axis and the function $\operatorname{f}_{n}$: Show that the series $\sum_{1}^{\infty}A_{n}$ ...
MiguelCG's user avatar
  • 339
0 votes
1 answer
27 views

Upper bound of $\int_{0}^{1}x^{2n-2} (1+x)^{n} e^{-cx^2}{_2F_1}(-\frac{n}{2},n-1;n-0.5;x)dx$

The integral is $\int_{0}^{1}x^{2n-2} (1+x)^{n} e^{-cx^2}{_2F_1}(-\frac{n}{2},n-1;n-0.5;x)dx$, where $n$ is a positive integer, $c$ is a positive real number. Is there a closed-form result of this ...
jobs adam's user avatar
0 votes
0 answers
27 views

How to Identify the General Function from a Given Recurrence Relation?

I am currently working on a problem involving graphs, and I have derived a recurrence relation as part of my work. However, I am struggling to identify the general function that solves this recurrence ...
Mohammad reza Khaniha's user avatar
0 votes
0 answers
9 views

Is the non-uniform membership problem for sets obtained from 1 by applying affine integer functions P/NP-complete/other?

The question concerns algorithmic complexity of the membership problem for sets obtained from $1$ by applying a fixed number of affine functions. One such example is Klarner-Rado sequence (A002977 in ...
lolicomu's user avatar
0 votes
0 answers
42 views

Pointwise convergence of $\frac{x^n}{1+x^n}$ [duplicate]

It is a question of pointwise convergence and I want to know whether I did it in the correct way or not. $\lim\limits_{x \to \infty} \frac{x^n}{1+x^n}$ By dividing with $x^n$, i got $\lim\limits_{x \...
TinCan's user avatar
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