The sequences-and-series tag has no wiki summary.

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**1**answer

302 views

### Computing the limit of a certain recursively defined sequence

The following is not exactly a research question (it was originated from manufacturing of exercises for calculus), and has no other motivation than explaining a phenomenon. I apologize if it is ...

**2**

votes

**0**answers

135 views

### Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture
It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...

**14**

votes

**2**answers

1k views

### The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...

**0**

votes

**1**answer

199 views

### Good set theory in which to study ordinal-indexed sequences?

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...

**1**

vote

**1**answer

183 views

### On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...

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votes

**2**answers

247 views

### sequences - recurrence relation [closed]

I have to find the expression of $(y_n)$ defined by :
$$y_{n+1}=a y_n+b z_n+c$$
where $(z_n)$ is an arithmetico-geometric sequence :
$$z_{n+1}=d z_n+e$$
and $a,b,c,d,e$ real numbers.
Thank you ...

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votes

**0**answers

227 views

### Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...

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votes

**4**answers

810 views

### Another Chicken or Egg: Sequence or Series

This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...

**8**

votes

**4**answers

668 views

### What is the theoretical interest of finding closed-form sols. of infinite series?

Hi,
I was reading this when I came across Gourevitch's conjecture.
My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve ...

**5**

votes

**1**answer

402 views

### An infinite set of identities using Stirling numbers 1st kind - are they all zero?

I have the following set of series involving the Stirling numbers 1'st kind and binomials, which can be understood as a set of dot-products of row- and column-vectors of two infinite matrices (where R ...

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votes

**2**answers

295 views

### Evaluating a limit similar to the Euler constant

In the course of studying a certain complex-valued functional equation, I have had a need to evaluate the following limit:
$$\gamma_\mathcal{T}=\lim_{n\to\infty}\left(-\frac{i}{2}\sum_{k=1}^n ...

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votes

**0**answers

391 views

### Composition of two formal series

There are two formal semi-infinite Laurent series
$$
f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k
$$
and
$$
f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}
$$
Their composition (we assume that this composition ...

**0**

votes

**1**answer

198 views

### A series representation

How to find the end of a series representation of the product
$$
\prod_{\substack{i=1...\infty\\\ j=0...i\\\ k=0...j}}\frac{1}{1-x^{i-j}y^{j-k}z^{k}}?
$$
For example for product
$$
...

**1**

vote

**1**answer

291 views

### Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence

Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.
Suppose I am drawing k numbers out of n numbers labeled from 1 to n.
Considering all ...

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votes

**1**answer

126 views

### what this type of series expansion is

I obtained a series expansions as this type
$$f(x)=g(x)^{\textstyle \sum_{i=0}^{n}\alpha_{i}x^{-i}+O\left(\tfrac{1}{x^{n+1}}\right)}$$
what is the exact name of this formula

**1**

vote

**1**answer

191 views

### Growth of a particular sequence

Not sure if this is research level? While analyzing a particular algorithm, I came across the following series:
$\sum_{k=1}^{m-1}\frac{3^{k}}{\prod_{i=1}^{k-1}2^{2^{i}}}$
Is there a closed form ...

**1**

vote

**2**answers

520 views

### Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left ...

**4**

votes

**0**answers

217 views

### $\sum_{p,q \text{ primes } p \le q} 1/(pq\log(pq))$

The sum $$ \sum\limits_{p,q \text{ primes } p \le q} \frac{1}{pq\log(pq)}$$
is related to a conjecture of Erdős about primitive sequences.
It converges because the sequence is primitive. If my ...

**1**

vote

**1**answer

129 views

### expanding the sqare of sum

If there any way to expand the following?
$$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$
and more generally, a way to expand
$$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$
where $gcd(p,q) = 1$
...

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vote

**2**answers

491 views

### Converting a recursive definition to an explicit one

Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$?
I've listed out the first few terms:
for $x=0,1,2,3,4,5,6, 7$
we have $a_x ...

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vote

**1**answer

395 views

### Generalization of Cauchy product

Hello,
working on some machine learning problem I end up facing a problem which looks like generalizing the notion of Cauchy product.
I briefly go back to Cauchy products before exposing my ...

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votes

**2**answers

328 views

### Equidecomposable graphs, unimodality and asymptotics

I will call two graphs $G$ and $H$, $r$-equidecomposable (in analogy with Hilbert's third problem) if they can be written as unions of disjoint subgraphs
$$G\cong \bigsqcup_{i=1}^r G_i\quad ,\quad ...

**9**

votes

**1**answer

451 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 $(x_0, x_1, \dots)$ ...

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votes

**3**answers

535 views

### The digit sum: $s(na)=s(nb)$

Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post.
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal ...

**4**

votes

**0**answers

247 views

### Useful lower bound on an infinite sum

Fix integer $s.$ I have encountered the following infinite sum.
$$\sum_{k=0}^\infty \Pi_{l=1}^k\left[1-(1-2^{-l})^s\right]$$
Is there a useful lower bound on this expression? For instance, if $s=1,$ ...

**4**

votes

**1**answer

2k views

### sum calculation

I would like to calculate, or bound from above, the following sum
$$
\sum_{i=0}^n(n-2i)^p{p \choose i},
$$
here $p\geq 2$.
Any references are very welcome.
Thank you.

**1**

vote

**1**answer

349 views

### Cesaro means for $\alpha<1$ and Banach limits

I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all almost convergent sequences. For the Cesaro summation method ...

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**2**answers

1k views

### Cesaro means and Banach limits

Consider the class of bounded sequences to which every Banach limit (non-negative shift-invariant continuous functional on $l^\infty$ taking convergent sequences in the usual sense to their limits) ...

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**1**answer

126 views

### An operator realizing the Borel transform

Let $y(z) = \sum_k y_k z^k$ be a holomorphic function in a vicinity of the point $z=0$. Define its Borel transform $By$ as a function $By(z) = \sum_k \frac {y_k}{k!} z^k$.
The well-know formula ...

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**0**answers

500 views

### “closed form” finite sum

If a finite sum has a definite integral representation, for which it can be proved the underlying indefinite integral is not an elementary function, then does this imply the original finite sum can ...

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**1**answer

318 views

### Is it true that $c_0(X)^* = \ell_1(X^*)$ ?

I'm trying to prove this that but I can't . Any help/reference ?

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**0**answers

199 views

### Proving that an increasing iterative sequence increases at a decreasing rate

In this question
Proving a sequence of integrals increases (iterated minimax distributions)
Pietro Majer proved that
$$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...

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votes

**2**answers

319 views

### Finding local patterns in a circular list

Consider a list $\boldsymbol{x}=x_0,x_1,\ldots,x_{n-1}$, which we consider to be circular by taking the subscripts modulo $n$. The entries in the list are distinct integers.
A local pattern is a ...

**0**

votes

**1**answer

418 views

### Sequence of smooth functions converging to sgn(x)

I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property:
\begin{equation}
f_i(x_1+x_2) = g_i(x_1, f_i(x_2))
...

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votes

**3**answers

780 views

### Simplifying finite sum over 1/(ax+b)

Can I simplify:
\begin{equation}
\sum_{x=x_0}^{x_1} \frac{1}{ax+b}
\end{equation}

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vote

**1**answer

188 views

### About one series. Are there some related special functions?

Hello,
I have the following series:
$$
\sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0,
$$
where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...

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votes

**2**answers

509 views

### The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$

Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...

**5**

votes

**1**answer

535 views

### random hyperharmonic series

The Harmonic Series is defined as:
$\sum_{n} \frac{1}{n}$ where $n=1,2,3,4....$.
This series is known to be divergent.
A generalization of this series can be made by raising each term to $p$:
...

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votes

**0**answers

222 views

### Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely
$$ \small f_p(x) = \sum_{k=0}^{\infty} ...

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**0**answers

219 views

### Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ contains arbitrary long arithmetic progressions?

This is somewhat related to Erdős conjecture on arithmetic progressions
Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ ...

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**2**answers

423 views

### Alternating sums of GCDs

The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be ...

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votes

**2**answers

3k views

### Alternating sum of square roots of binomial coefficients

Let
$$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$
It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...

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votes

**7**answers

5k views

### Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.) If so, how would one prove this? (To confirm: This is the power
series for $e^x$, except with the ...

**3**

votes

**1**answer

337 views

### Products of trigonometric functions with increasing frequencies

I am looking at weighted $L_2$ norms of a class of Littlewood polynomials, related to Walsh and Rademacher functions which made me look for pseudo-closed forms or computationally efficient expressions ...

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votes

**1**answer

421 views

### Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense.
OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3]
Numerical evidence suggest that up to $10^5$
$$ ...

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**0**answers

844 views

### Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...

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**2**answers

825 views

### Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...

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**0**answers

776 views

### Has n^2*|sin(n)| limit? [closed]

Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity.
In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...

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votes

**2**answers

2k views

### Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...

**2**

votes

**2**answers

1k 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 ...