All Questions
Tagged with real-analysis sequences-and-series
304 questions
5
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Weaker versions of the Riemann series theorem in constructive mathematics
The classical Riemann series theorem states that given a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all real ...
- 2,624
5
votes
1
answer
630
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Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
- 10.5k
5
votes
0
answers
67
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Minimal growth condition for a rearrangement
Let $\sigma: \mathbb{N}\to\mathbb{N}$ be bijective such that there is a sequence $(n_k)_{k\ge 0}$ in $\mathbb{N}$ satisfying $|\sigma(n_k)−n_k|\to\infty$ for $k\to\infty$.
Question: Is there a (...
- 2,160
5
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343
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Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
4
votes
1
answer
1k
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seeking proofs: infinite series inequalities
Question. Numerically, the following is convincing. However, is there a proof?
$$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4
<\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\...
- 43.2k
4
votes
2
answers
415
views
The set of all possible values of subseries of a convergent positive term series
Inspired by The set of all limits of sub-series of an absolute convergent series is the following true?:
Let $a_n$ be a strictly decreasing sequence and $\sum_1^\infty a_n=\ell<\infty$ is a ...
- 356
4
votes
3
answers
2k
views
Dominated convergence theorem when the measure space also varies with $n$
Let $(f_n)_n:X \to \mathbb R$ be a sequence of measurable functions on a measurable space $X$ converging pointwise to a function $f:X \to \mathbb R$, and let $(\mu_n)_n$ be a sequence of finite ...
- 6,853
4
votes
2
answers
303
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Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, with $\operatorname{gd}^{-1}(z)$ the inverse Gudermannian
I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or ...
4
votes
2
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592
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From Zurab's integral representation for the Apéry's constant to almost impossible integrals
I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
4
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2
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323
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Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?
Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that
$$\lim_{n} ~a_{nn} = L_1, \qquad
\lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$
exist. Is there a sufficient criteria to conclude ...
- 3,245
4
votes
2
answers
548
views
Convergence of a sequence
Let $x_0=1$ and
$$x_{k+1} = (1-a_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x_k}\right)$$
where $a_n$ is a known sequence satisfying that $a_k\in(0,1)$ for all $k$ and $a_k\to 0$ as $k\to\infty$. How to ...
- 311
4
votes
1
answer
581
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A question on null sequences
Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero?
In case the answer is affirmative (...
- 13.5k
4
votes
2
answers
5k
views
Ratio of Sequences Sum Inequality
I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
- 1,182
4
votes
1
answer
203
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An infinite series involving Jordan's totient function
Can you provide a proof for the following claim:
$$-\displaystyle\sum_{n=1}^{\infty}\frac{J_k(n)}{n} \cdot \ln\left(1-x^n\right)=\frac{x \cdot A_{k-1}(x)}{(1-x)^k} \quad \text{for} \quad |x| < 1 \...
- 2,661
4
votes
1
answer
293
views
Points of differentiability of $f(x) = \sum\limits_{n : q_n < x} c_n$
Let, $\{q_n\}_{n \in \mathbb{N}}$ be an enumeration of rational numbers. Consider the function $f : \mathbb{R} \to \mathbb{R}$ given by, $$\displaystyle f(x) = \sum\limits_{n : q_n < x} c_n$$
...
- 810
4
votes
1
answer
155
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Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous?
Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate ...
- 307
4
votes
1
answer
129
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Convergence of a real sequence (stochastic approximation)
Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$
I would like ...
- 113
4
votes
1
answer
214
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Is this set of sum over a diagram is uncountable set?
This problem has been posted in Mathstack but number of responses is very low (a answer is given but does not look correct).
Consider the following diagram:
Let $0<x<\frac{1}{2}$
Note that $[.]$...
- 53
4
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1
answer
254
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$\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$
Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$
Compute $f(1)$ and $f(2)$.
4
votes
1
answer
188
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Is there a generalization of these q-series identities?
Denote $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$.
The below three identities are known.
\begin{align*}
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(q;q)_n}
&=1-\sum_{n\in\mathbb{Z}}(-1)^nq^{\...
- 43.2k
4
votes
1
answer
693
views
An asymptotic expansion of a infinite sum
I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula ...
- 193
4
votes
1
answer
2k
views
Kolmogorov tightness criterion for stochastic processes
I am searching for the criterion stated above and also here: The question about Kolmogorov tightness criterion.
It should state the following: If a sequence of stochastic processes $(X^n)$ fulfills:
...
- 43
4
votes
2
answers
225
views
Extremal problem for sequences
Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...
- 49
4
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1
answer
317
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Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
- 73
4
votes
1
answer
339
views
Limit for series of Bessel functions evaluated at zeros
The following series arises in an electrostatics problem for a conducting cylinder:
$$
V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}
$$
where $J_i$ is the Bessel function of $i^{th}$ ...
- 573
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 284
4
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1
answer
119
views
Proving Equal Set Sizes in Sequential Point Selection on a Real Interval with Variable-Length Intervals
I'm here as an engineer working on a point sampling algorithm and I've noticed that when I perform the algorithm on an ordered set of points in one direction it selects the exact same number of points ...
- 151
4
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0
answers
459
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Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?
Is there any closed form expression for the following serie?
$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$
Or at least a proof that it is an irrational number. The ...
3
votes
1
answer
572
views
Remarkable limit involving $m_p=\log_p(p^{x_1} + \cdots + p^{x_n})-\log_p(n)$
It is easy to prove that
$\lim_{p\rightarrow 1} m_p = (x_1 + \cdots + x_n)/n$. The following fact about the derivative of $m_p$ with respect to $p$ is also elementary:
$$m'_p =\frac{dm_p}{dp}
=\frac{1}...
- 3,098
3
votes
1
answer
2k
views
A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm
Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...
- 11.9k
3
votes
1
answer
702
views
$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
- 3,098
3
votes
1
answer
314
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Product of sequences "slowly" converging to $0$
The starting point of this question is that $\sum_{n=1}^\infty \frac{1}{n^{\alpha}} < \infty$ if and only if $\alpha > 1$.
Let $(a_n)_{n\in\mathbb{N}}$ be a non-negative sequence. We say that $(...
- 48.9k
3
votes
1
answer
398
views
Constant "periodization" of a function
Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that
$$ \sum_{n \in \mathbb{Z}} w(x+n) = 0$$
for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume ...
- 9,853
3
votes
1
answer
631
views
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
- 128k
3
votes
1
answer
367
views
Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros
Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum_{...
- 219
3
votes
1
answer
594
views
What is the rate of convergence? [closed]
How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
- 11.3k
3
votes
1
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496
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Prove that these two definitions of "natural" integration constant coincide when both converge
These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...
- 10.1k
3
votes
1
answer
211
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Other expansion for positive Taylor expansion
I was thinking of the following problem. Let $f$ be a Taylor expansion and $a_k$ the associated coefficients,
$$\forall x\in\mathbb{R},~f(x)\triangleq\sum_{k=0}^\infty a_kx^k.$$
Let suppose that we ...
- 393
3
votes
1
answer
315
views
Where to find the proof of this property?
I am doing some exercises in the analytic and there is a problem as following:
``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that:
$\sum\limits_{n=1}^{+\infty} f_n = 1$.
$\...
- 153
3
votes
1
answer
133
views
A problem with sequences with composition of $\log$s
If $(a_n)_{n \ge 1}$ is a non-negative sequence s.t., $$\sum\limits_{n = c_k}^\infty \frac{a_n}{\log^{(k)} n} < \infty, \, \forall k \ge 1 \overset{?}{\implies} \sum\limits_{n \ge 1} a_n < \...
- 810
3
votes
2
answers
373
views
Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$
This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question
Let tangents be drawn to the curve $y=\sin x$ from ...
- 139
3
votes
1
answer
168
views
Uniform bound for an oscillatory sum
I am wondering if there is a uniform bound $C$ (independent of $\lambda>10$):
$$\sum_{k=-\infty}^{-1}\Big|\int_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\le C.$$
Remark: (1) An easy upper ...
- 187
3
votes
1
answer
331
views
Solving recurrent relation
I have the following recurrent relation and I want to find a close form of it if it exists at all.
$$
P_n = (1-p)^{n-1}P_{n-1} + \sum\limits_{k=2}^{n} \binom{n-1}{k-1} p^{\binom{k}{2}} (1-p)^{k(n-k)} ...
- 342
3
votes
1
answer
344
views
Asymptotic behavior of a recursion
Let $x_n(0)=1$,
$$
x_n(N+1) = \frac{1}{N+1}\sum_{k=0}^N \sum_{j=1}^n x_j(k)x_{n+1-j}(N-k) + \frac{10}{N+1} x_{n+1}(N) , \quad\quad N\ge 0 .
$$
So the recursion is on $N$, and at each level, we compute ...
- 24.2k
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
3
votes
1
answer
166
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,689
3
votes
1
answer
495
views
Simple closed forms for sums such as $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{qk - p}$ and related integrals
My goal here is to get a simple expression for $\zeta(3)$. This is a follow up to my previous question posted here. Any Taylor-like expansion from everything I tried won't make it. So this is my last ...
- 3,098
3
votes
1
answer
334
views
Does this function have any exponential growth?
Has anyone seen any function of the following type?
$$
g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant $c>...
- 1,649
3
votes
1
answer
248
views
"Lagrange inversion" around an extremum
Cross-posted from Math Stackexchange.
In an older question to which I provided an answer it was asked how to compute a particular limit involving the roots of a transcedental function around its ...
- 151
3
votes
1
answer
201
views
"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients
In a lecture I once attended, I remember the speaker using a result of the following nature:
$``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form
$$P(n) ...
- 739