Articles of sequences and series

Divergence of Reciprocal of Sequence knowing the Asymptotic Density

In a paper, it is stated that if a particular sequence of positive integers $a_1<a_2<\dots$ satisfies that the number of $a_i$ which do not exceed $n$ is $o(\frac{n}{\log ^2(n)})$, then $\sum \frac{1}{a_i}$ converges. I cannot see why this would be true- it’s probably really simple but it’s evading me right now. Help would be appreciated.

Is $\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$ right?

How to prove $$\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$$ I met this function when I tried to give another proof of the known lower bound of Tur\’an functions of complete hypergraphs ( Based on a same construction, instead of using shifting method, I tried to count edges directly ) Here is the question: Define $a_0=-1$ and $a_1=1$. For all $r\geqslant2$, […]

How do we find the $n+2$th term of the series $1 + (3+5) + (7+9+11)+\dots$?

We have the series $1 + (3+5) + (7+9+11)+\dots$. We need to find the $n+2$th term and hence summation of the series up to this term. However hard we try we do not seem to be able to fit this series to a pattern.

Absolute convergence of double series

I read two theorems which both involve absolute convergence. However, I am confused whether they are equivalent of if one implies the other. The first theorem is: If $\sum_{i,j=1}^{\infty} |a_{ij}|$ converges then $\sum_{i,j=1}^{\infty} a_{ij}$ converges. and the second theorem is: If the iterated series $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} |a_{ij}|$ converges (meaning that for each fixed $i \in \mathbf{N}$ […]

When is the first case where $67$ is the least prime factor of f(n)?

Let f(n)=$1234567891011$….n (concatenation of first n natural numbers). I make a sequence of numbers made with this following definition: Smallest number n such that the m-th prime number is the least prime factor/divisor of f(n). And I found these following: n=2 is the first case where 2 is the least prime factor of f(n) n=3 […]

Is it possible to identify this sequence?

Interested by this question, $j$ being a positive integer, I tried to work the asymptotics of $$S^{(j)}_n=\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+j)}=\frac{\, _2F_1\left(j,-n;j+1;-\frac{1}{n}\right)}{j}$$ I quickly noticed (not a proof) that the asymptotics write $$S^{(j)}_n=(-1)^j\left(\left(\alpha_0-\beta_0e\right)-\frac{\left(\alpha_1-\beta_1e\right)}{2n}+\frac{\left(\alpha_2-\beta_2e\right)}{24n^2}\right)+O\left(\frac{1}{n^3}\right)$$ in which the $\alpha_k$’s and $\beta_k$’s are all positive whole numbers depending on $j$. What I found is that $$\alpha_0=(j-1)!\qquad \qquad \beta_0=\text{Subfactorial}[j-1]$$ $$\alpha_1=(j+1)!\qquad \qquad \beta_1=\text{Subfactorial}[j+1]$$ $$\alpha_2=(1+3j)(j+2)!$$ […]

Conceptual basis for formula for sum of n first integers raised to power k,

Most programmers (including me) are painfully aware of quadratic behavior resulting from a loop that internally performs 1, 2, 3, 4, 5 and so on operations per iteration, $$\sum_{i=1}^n i = \frac{n \left(n+1\right)}{2} $$ It’s very easy to derive, e.g. considering the double of the sum like $(1+2+3) + (3+2+1) = 3\times4$ . For the […]

About the positive sequence $a_{n+2} = \sqrt{a_{n+1}} + \sqrt{a_n}$

Given the positive sequence $a_{n+2} = \sqrt{a_{n+1}}+ \sqrt{a_n}$, I want to prove these. 1) $|a_{n+2}| > 1 $ for sufficiently large $n \ge N$. 2) Let $b_{n} = |a_{n} – 4|$. Show that $b_{n+2} < (b_{n+1} + b_{n})/3$ for $n \ge N$. 3) Prove that the sequence converges. How should I proceed? Is there a […]

Matrix sequence convergence vs. matrix power series convergence:

Is my thinking correct? The sequence $A^n$ converges if each entry converges to a finite number. But for a matrix power series, $ I + A + \cdots + A^n + \cdots $ can never converge if it has, for example a “1” in the upper left corner, in entry $a_{11}$. Take, for simplicty, $A$ […]

Weak and strong convergence of sequence of linear functionals

Is this sequence of linear functionals weakly (strongly) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_2\,?$$