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-4
votes
0answers
42 views

Mathametical Proof for a series [on hold]

Prove for the series given below z/((1−z)^2) =∑N≥1 N(z^N) Do i need to do this by induction?
0
votes
2answers
69 views

Hypergeometric sum specific value

How to show? $${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$ It numerically is very close, came up when evaluating: $$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...
1
vote
2answers
251 views

How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$?

Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a strongly additive function on positive integer number $m$, where $p$ is a prime number. Set $${f_x}(p) = \left\{ ...
-2
votes
0answers
26 views

Summation Value of a Combination Function [closed]

sum of 5^n / (n! * 2^n) as n goes from 1 to infinity. I know how to prove that the above series is convergent but I'm not sure how to get the value that it converges to. If it were just the ...
0
votes
0answers
23 views

Regarding to $ \lim_{N\rightarrow \infty}\sum_{n=1}^{N+1}\sum_{k=n-1}^{N}C_{kn}$, prove or disprove! [migrated]

Regarding to $$ \lim_{N\rightarrow \infty}\sum_{n=1}^{N+1}\sum_{k=n-1}^{N}C_{kn}, $$ prove or disprove: (a) If $\sum_{k=n-1}^{\infty}C_{kn}$ is convergent to $c_n$, for all $n\geq 1$, and $ ...
0
votes
0answers
16 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
0
votes
0answers
105 views

Base-signed harmonic series

This question is directly related to this question by Douglas Zare. The harmonic sequence $$s = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}+\frac{1}{7}-\frac{1}{8} - ... ...
1
vote
1answer
129 views

Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional Morse Theory and Multiple Solution Problems": if i have then $(4.1)$ is formal : it means that EDIT1: $(4.1)$ tel us that ...
15
votes
1answer
489 views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$, define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) ...
1
vote
2answers
191 views

Looking for a limit related to the series in a previous post

Can any one show that the following limit? $$ \lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1. $$ If one uses the ...
2
votes
2answers
292 views

How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration? $$ \int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0. $$ This post is related to my previous question here , ...
17
votes
3answers
2k views

Where in mathematics do these polynomials appear?

Does anyone recognize the following sequence of polynomials? $f_0(x) = x-1$ $f_1(x) = x^2-x$ $f_2(x) = x^4-2x^2+x$ $f_3(x) = x^8-3x^4+3x^2-x$ $f_4(x) = x^{16}-4x^8+6x^4-4x^2+x$ $\vdots$ The ...
-1
votes
1answer
235 views

Does anyone recognize this generating function [closed]

$a_1=1, a_2=1, a_3=3, a_4=15, a_5=105$ Reccurence formula is $a_{k+1}=\sum\limits_{\lambda_1+\lambda_2+\ldots+\lambda_s=k,\ \lambda_i\geq1} a_{\lambda_1}a_{\lambda_2}...a_{\lambda_s}{k \choose ...
7
votes
1answer
487 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
vote
0answers
93 views

Passing to the limit in a PDE (subsequence problems)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
-2
votes
1answer
77 views

Monotonic sequence (edited) [closed]

For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v_i\leq v_i'$. Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a_1, ...
3
votes
0answers
85 views

“Shifted” Vandermonde determinant is nonzero?

I have already posted this question at MSE here, but as it received a few upvotes, but no comments or answers I choose to cross-post it here. Let $P$ be a degree-two polynomial, with roots ...
0
votes
1answer
65 views

Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$ such that this sequence has uniform ...
2
votes
3answers
289 views

Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ? $$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...
2
votes
1answer
164 views

Collatz dropping times aperiodicity

Let $a(n)$ for $n \geq 1$ be the number of powers of two $2^m$ that lie between $3^{n-1}$ and $3^{n}$: $$a(n) = 1, 2, 1, 2, 1, 2, 2, 1, ...$$ It represents the increments between successive terms of ...
5
votes
0answers
142 views

Irrationality of the sum of the reciprocal of perfect powers

A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that ...
0
votes
0answers
36 views

Supersets of P-finite sequences and rings

P-finite sequences are a superset of C-finite sequences. While doing programming work, the question came up what generalizations or supersets of P-finite sequences have people described. In other ...
3
votes
0answers
101 views

Prove that when converge, the following expansions are equal

Prove $f_1(x)=f_2(x)=f_3(x)$ when converge. $$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)$$ $$f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom ...
2
votes
1answer
98 views

a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation: $$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$ where $A>B>0$. This equation is related to a real polynomial (see here) which I want to ...
6
votes
0answers
163 views

Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
0
votes
0answers
64 views

Usage of multinomial theorem with infinite series

What are the conditions for using the multinomial theorem with infinite series? I have an expression but I don't know if I can use it. The expression is: $$ \left[\sum_{m=0}^{\infty} \frac{\mu^m ...
7
votes
3answers
288 views

Summation of a series

I would like to sum the series $$ \sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} . $$ It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...
0
votes
1answer
68 views

Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define $$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...
16
votes
1answer
108 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
3
votes
2answers
107 views

series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
1
vote
0answers
128 views

Asymptotic behaviour of sequence

I am interested in the sequence $$a(n)=\sum_{k=0}^n {p(n-k)-1 \choose k}$$ where $p(n)=(r-1)n^2+(2r-1)n+r$ for some $r \in \mathbb{N}$ or more generally any polynomial equation. When $r=1$ this ...
3
votes
1answer
232 views

Series of the inverse quadratic trinomial

Maybe it's a very simple question, but I have a problem with the following series $$\sum\limits_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$ where $p, q \in \mathbb{R}$. I know about five ways how to calculate ...
3
votes
2answers
245 views

norm of the matrix series

The goal is to obtain an upper bound for the norm of the vector $$ \left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\| $$ for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ...
10
votes
1answer
666 views

Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made: Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened. Define ...
4
votes
1answer
144 views

Extending a Certain Result from Locally Convex Topological Vector Spaces to General Topological Vector Spaces

In this Math Stack Exchange post, I proved the following result. Theorem: Let $ X $ be a locally convex topological vector space. Let $ x \in X $ and suppose that $ (x_{n})_{n \in \mathbb{N}} $ is ...
1
vote
0answers
267 views

Do these infinite series expressing $\zeta(s)$ only (partially) converge at $\Re(s)=\frac12$?

The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ was derived here: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(\sum _{n=1}^{\infty } {\frac {s-1-2\,n}{{n}^{s}}} + ...
16
votes
3answers
1k views

Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question): The Riddle: We assume ...
0
votes
2answers
155 views

Finding the min of a sequence related with factorials

Let $N,n$ be natural numbers. Let us define $a_n=m$ when $N!$ can be divided by $(n!)^m$ and it cannot be divided by $(n!)^{m+1}$. For a given $N(\ge 2)$, let $\min(N)$ be the min of $na_n\ (2\le ...
0
votes
0answers
94 views

Multivariate generating function

I am investigating the perturbation of the Jordan canonical form. In my work I must calculate the number of ways to factor $p^ {n-k} q^k$ where $p$ and $q$ are distinct primes ...
10
votes
1answer
491 views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{p}} $$ Additional information: Since $$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{n}} ...
3
votes
1answer
133 views

Infinite series - analytical solution

Analytical Solution is required for: $$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty ...
7
votes
1answer
274 views

Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$

I am currently interested in the following sequence: $$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the ...
0
votes
0answers
63 views

Proof that Newton expansion over derivatives has the properties of an integral [duplicate]

Let's consider a Newton expansion over consecutive derivatives of a function: $$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ Can it be proven that such ...
0
votes
0answers
29 views

Implications of natural functions (as defined here) to integrals and iterations

This is a split from the previous question which I re-formulated to better match the received answer. Let's define a natural function as a continuous function that is equal to its Newton expansion: ...
2
votes
1answer
181 views

How to find the coefficients of a poor-converging series?

I have the series $\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$ and the boundary conditions $\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...
1
vote
0answers
73 views

A generalization of alternating series involving modulus?

Alternating series are common in the literature, with important examples including $\displaystyle\sum_{n=1}\frac{(-1)^{n-1}}{n}=\log 2$, ...
7
votes
1answer
380 views

Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping $$ f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n} $$ surjective? If not, what is its image? If yes, what can be said about ...
5
votes
0answers
118 views

Inverse problems for an asymptotic series which depends on a parameter?

I have the series $\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$, where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An ...
1
vote
0answers
85 views

Structural differences between closed forms of two related infinite products?

In this question, I went quite a bit over the top, so I now tried to rephrase it in a much simpler way. Take $a \in \mathbb{R}, s \in \mathbb{C}$ and: $$\displaystyle C(s,a) := \prod_{n=1}^\infty ...
3
votes
0answers
488 views

$\sum_{n=1}^{\infty}\frac{1}{a_n}=\infty$ $\sum_{n=1}^{\infty}\frac{1}{b_n}=\infty$ but $\sum_{n=1}^{\infty}\frac{1}{a_n+b_n}=c, c\in R$ [closed]

The following question is inspired from: Defining the slowest divergent series. Let $a_n$ and $b_n$ be two strictly increasing sequences of natural numbers,with ...