All Questions
Tagged with real-analysis sequences-and-series
304 questions
122
votes
5
answers
27k
views
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 4,535
54
votes
3
answers
4k
views
Does every real function have this weak continuity property?
In my research I came across the following question :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
- 4,074
46
votes
2
answers
7k
views
Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
- 573
25
votes
3
answers
2k
views
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits_{n\to\infty}\prod\limits_{k=1}^n f(\frac{k}{n})<\infty$ ?
I do not see any reason why such a function could ...
- 3,527
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
21
votes
3
answers
1k
views
What is the set of all "pseudo-rational" numbers (see details)?
Define a “pseudo-rational” number to be a real number $q$ that can be written as
$q=\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$
Where $P(x)$ and $Q(x)$ are fixed integer polynomials (independent of n). ...
- 311
18
votes
5
answers
3k
views
Bernoulli sum meets golden number
Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio.
I encountered the following infinite sum and would like to ask:
Question. Is this true? If so, any ...
- 43.2k
17
votes
2
answers
2k
views
"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?
Recently, I encountered this problem:
"Given a sequence of positive number $(x_n)$ such that for all $n$,
$$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$
Find the limit $\lim_{n \rightarrow \infty} \...
- 327
16
votes
1
answer
661
views
Does every real function have this weak derivation property?
After this question : Does every real function have this weak continuity property?
Natrualy there are an other (more difficult) :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
- 4,074
15
votes
4
answers
3k
views
No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
15
votes
3
answers
2k
views
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I've been trying to find an asymptotic expansion of the following series
$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$
and
$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
- 153
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
15
votes
3
answers
903
views
Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $
I am trying to prove or disprove
$$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$
where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
- 5,474
14
votes
1
answer
481
views
A question on a real sequence
Let $\{a_n\}_{n\ge1}$ be a real sequence that decays faster than any algebraic speed, that is, $\lim_{n\to \infty} n^pa_n = 0$ for every positive integer $p$. Assume that $$\sum_{n\ge 1}(n+1)^kn^ka_n =...
- 903
14
votes
1
answer
900
views
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
Volterra ...
- 4,897
13
votes
3
answers
1k
views
iterated harmonic numbers vs Riemann zeta
Define the $m$-th iterated harmonic sums in the manner: $\bar{H}_0(n):=1$ and for
$m\geq1$ by
$$\bar{H}_m(n):=\sum_{k=1}^n\frac{\bar{H}_{m-1}(k)}k.$$
For example, $\bar{H}_1(n)=\sum_{k=1}^n\frac1k$ ...
- 43.2k
13
votes
3
answers
720
views
Supremum of $ a_n = a_{n-1}^3 - a_{n-2} $
Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $
Define
$$ a_n = a_{n-1}^3 - a_{n-2} $$
Then
$$ \sup_{n>2} a_n = a_2 $$
And
$$ \inf_{n>2} a_n = - a_2 $$
How to prove that ?
- 763
13
votes
2
answers
1k
views
Is there a known condition for partial sums of a decreasing positive sequence to take all values up to the total sum?
Let $a_0>a_1>\cdots>0$ have the property that, for each positive $a<\sum_{n\in\Bbb N}a_n$ (admitting $\infty$ for the sum), there is $A\subset\Bbb N$ such that $a=\sum_{n\in A}a_n$ . Are ...
- 2,437
13
votes
2
answers
1k
views
On Hamkins' answer to a problem by Michael Hardy
Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum $\mathfrak{...
- 2,381
13
votes
3
answers
810
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
12
votes
2
answers
663
views
A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$
I am looking for a proof of the following claim:
First define the function $\chi(n)$ as follows:
$$\chi(n)=\begin{cases}1, & \text{if }n \equiv \pm 1 \pmod{10} \\
-1, & \text{if }n \equiv \pm ...
- 2,661
12
votes
1
answer
765
views
Possible limit involving the gamma function
Does $$\lim_{n \to \infty} \int_{0}^{1} \Gamma(x)^{n/(n+1)}dx - n$$ exist?
Here's some background. The integral
$$\int_{0}^{1} \Gamma(x) dx$$
diverges rather slowly. Inserting the exponent $n/(n+1)$ ...
- 3,443
12
votes
1
answer
742
views
If the generating function summation and zeta regularized sum of a divergent series exist, do they always coincide?
One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
- 4,829
12
votes
1
answer
1k
views
Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are "sort of increasing" or "sort of decreasing" (as defined below)?
Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...
- 167
11
votes
2
answers
1k
views
Two divergent series conspiring?
Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus
$$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad
\sum_{n\geq0}\frac{a_n}{2n+...
- 43.2k
11
votes
1
answer
1k
views
Has anyone seen this series?
I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...
- 1,649
11
votes
2
answers
587
views
Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
- 271
11
votes
1
answer
1k
views
Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$?
The series $\sum_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)...
- 462
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 113
11
votes
0
answers
322
views
Does any real function have a Lipschitzian restriction on $D$?
Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
- 4,074
10
votes
4
answers
1k
views
Adventure with infinite series, a curiosity
It is easily verifiable that
$$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
...
- 43.2k
10
votes
2
answers
886
views
An attempt to generalize the previous inequality
In my previous MO question, the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps ...
- 43.2k
10
votes
2
answers
1k
views
Does a conditionally convergent sum with random signs converge almost surely?
Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
- 6,215
10
votes
2
answers
597
views
How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$
I'm generally interested in being able to find an asymptotic expansion of
$$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$
As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...
- 2,689
9
votes
2
answers
2k
views
An inequality involving square roots and sums
I've been trying to prove (maybe even disprove) the following inequality:
$$
\sum_{n=1}^{N} \frac{a_n}{\sqrt{\sum_{i=1}^{n}a_i}} \leq C \sqrt{\sum_{n=1}^{N}a_n}
$$
Where $ a_1,...,a_N\geq 0 $ are ...
- 109
9
votes
1
answer
692
views
An infinite series involving harmonic numbers
I am looking for a proof of the following claim:
Let $H_n$ be the nth harmonic number. Then,
$$\frac{\pi^2}{12}=\ln^22+\displaystyle\sum_{n=1}^{\infty}\frac{H_n}{n(n+1) \cdot 2^n}$$
The SageMath ...
- 2,661
9
votes
2
answers
354
views
Asymptotics of a quadratic recursion
Consider the sequence defined by
\begin{align}
c_0 &{}= 1 \\
c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}.
\end{align}
How can you prove that it has the following asymptotics ...
9
votes
1
answer
553
views
Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
- 541
9
votes
1
answer
556
views
A non-recursive, explicit formula for the Fabius function
The Fabius function $F\colon\mathbb R\to[-1,1]$ may be defined as the unique solution of the functional integral equation
$F(x)=\int_0^{2x}F(t)\,dt$ for all real $x$ such that $F(1)=1$.
The recent ...
- 128k
9
votes
2
answers
490
views
Rearrangement, conditional convergence, and "placid" permutations
This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
- 20.9k
9
votes
3
answers
362
views
Maximum zero converges to $\sqrt{2}$
In my research I came upon a recursively defined sequence, and I'm pretty sure it converges to $\sqrt{2}$ though I can't prove it easily. I don't think it is a difficult question but I'm not sure.
...
- 342
9
votes
1
answer
845
views
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE 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$, ...
- 541
9
votes
2
answers
440
views
How to prove this sum involving powers of cosec is an integer?
It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$.
$F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...
- 201
9
votes
0
answers
180
views
Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
- 43.2k
8
votes
3
answers
1k
views
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found ...
- 2,661
8
votes
1
answer
1k
views
When can a sum be re-signed to converge to any limit?
Let $a_n$ be a sequence of positive real numbers with $\sum a_n < \infty$. What are the necessary and sufficient conditions for the following to hold?
For any $S \in \mathbb R$ with $-\sum a_n \...
- 6,215
8
votes
1
answer
552
views
Is $\sum_{k=0}^n (|\sin(k)|-2/\pi) $ bounded by a constant $M$?
I know $\sum_{k=0}^{n} \sin(k)$ is bounded by a constant
and $\sum_{k=0}^{n} \sin(k^2)$ is not bounded by a constant.
Then, what about $\sum_{k=0}^{n} (|\sin(k)|-2/\pi)$?
From numerical ...
- 275
8
votes
0
answers
518
views
Concave and other bounded functions: Series representation and converging polynomials
Main Question
Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
- 697
7
votes
2
answers
479
views
Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$
Let $f:\mathbb{R}\to\mathbb{R}$ be the function $$f(x) = \sum_{k=1}^{\infty} \sqrt{\max\left\{1 -\frac{k^2}{x^2},0\right\}}.$$
Numerical experiments suggests that there exists $n\in\mathbb{N}$ and a $...
- 2,605
7
votes
1
answer
348
views
Descartes' rule of signs for infinite series
Consider the function given by
$$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$
where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
- 43.2k