Articles of sequences and series

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where $_2F_1$ is the hypergeometric function and binomial $\binom n k$. The first few numerators $N$ are, $$N_1(n) = \color{brown}{1, \,5, \,43, \,177, \,2867, \,11531, 92479}, \,74069, 2371495,\dots$$ $$N_2(n) = \color{brown}{1, \,5, \,43, \,177, \,2867, \,11531, 92479}, \,370345, 11857475,\dots$$ respectively, and […]

Given the first $n$ terms of a sequence, can we find a function which would generate any infinite sequence we like that has the same first $n$ terms?

I was thinking about questions you sometimes see of the form Find the next $3$ terms of the sequence $$2, 3, 5, 7, …$$ Presumably this example would want us to find the next three prime numbers, but it occurred to me that this could also be the sequence of roots, in ascending order, of […]

Existence of a sequence with prescribed limit and satisfying a certain inequality II

This is a more specific variation of the question in the post Existence of a sequence with prescribed limit and satisfying a certain inequality Suppose you have two infinite sequences $\{a_n\},\{b_n\}$, with $0<a_n<b_n < 1$ for each $n$, such that both $a_n, b_n \to 1$ as $n \to \infty$. Further assume that $a_n/b_n \to 1$ […]

Wiedemann for solving sparse linear equation

I am new member. I am researching in Wiedemann algorithm to find solution $x$ of $$Ax=b$$ Firstly, I will show a Wiedemann’s deterministic algorithm (Algorithm 2 in paper Compute $A^ib$ for $i=0..2n-1$; n is szie of matrix A Set k=0 and $g_0(z)=1$ Set $u_{k+1}$ to be $k+1$st unit vector Extract from the result of step […]

Generating function for $\sum_{k\geq 1} H^{(k)}_n x^ k $

Is there a generating function for $$\tag{1}\sum_{k\geq 1} H^{(k)}_n x^ k $$ I know that $$\tag{2}\sum_{n\geq 1} H^{(k)}_n x^n= \frac{\operatorname{Li}_k(x)}{1-x} $$ But notice in (1) the fixed $n$.

$\sum_{n=1}^\infty a_n<\infty$ if and only if $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$

For $a_n$ positive sequence. I think I can prove one direction, but not both.

Clarification on how to prove polynomial representations exist for infinite series

With reference to this question, I would like a clarification of the comment given by @Ant (but someone else could answer instead). I basically have 2 questions: Is there any formal way to prove that there exists a polynomial representation for the sum of the first $n$ natural numbers to the $m^{th}$ power ($1^m+2^m+3^m+\cdots+n^m$) without […]

Fourier series for square-wave function?

Show that the Fourier series for the square wave function $$f(t)=\begin{cases}-1 & -\frac{T}{2}\leq t \lt 0, \\ +1 & \ \ \ \ 0 \leq t \lt \frac{T}{2}\end{cases}$$ is $$f(t)=\frac{4}{\pi}\left(\sin\left(\frac{2\pi t}{T}\right)+\frac{\sin(\frac{6\pi t}{T})}{3}+\frac{\sin(\frac{10\pi t}{T})}{5}+……\right)$$ I understand that the general Fourier series expansion of the function $f(t)$ is given by $$f(t)=\frac{a_0}{2}+ \sum_{r=1}^{r=\infty}\left(a_r\cos\left(\frac{2\pi r t}{T}\right)+b_r\sin\left(\frac{2\pi r t}{T}\right)\right)$$ But […]

Convergence of the series $\sum_{n = – \infty}^{\infty} (\sqrt{(a+n)^2+b^2} – |n| )$

Given $a, b \in \mathbb{R}$ with $b > 0$, is the series \begin{equation} \sum_{n = – \infty}^{\infty} (\sqrt{(a+n)^2+b^2} – |n| ) \end{equation} convergent or divergent? If we drop out the $n=0$ term and fold the remaining sum, the question can be equivalently asked for the series \begin{equation} \sum_{n = 1}^{\infty} (\sqrt{(a+n)^2+b^2} + \sqrt{(a-n)^2+b^2} – 2n […]

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of the Riemann integral. I want some hint or example, Im unable to find a sequence like this. My work at this moment: If the integral converges to […]