Articles of sequences and series

Series involving factorial

Let $$f(n,k) = \dfrac{1! + 2! + … + n!}{(n+k)!}$$ Find (if exists) the values of $k$ such that $\sum_{n=1}^{\infty} f(n,k)$ converges. What I’ve done so far: If exists $k$ such that $\sum_{n=1}^{\infty} f(n,k)$ converges, then $\sum_{n=1}^{\infty} f(n,M)$ converges for all $M\ge k$ That is because $f(n,k+1) = \dfrac{1}{n+k+1} f(n,k)$ and apply Abel’s Test. For […]

Dense subseries of divergent series

Suppose $\sum_{n>1} a_n=\infty$ and $0<a_{n+1}<a_n$. Let $b_k=a_k$ or $b_k=0$ for all integers $k$. Let $R=\lim_{n\rightarrow\infty}((1/n)\sum_{q=1}^{q=n} b_q/a_q)$ If $R>0$, how to show that $\sum_{n>1}b_n=\infty$? If $0<\lim_{n\rightarrow\infty}((1/\sqrt n)\sum_{q=1}^{q=n} b_q/a_q)$, must $\sum_{n>1}b_n=\infty$?

Convergence of a sequence of Random variables

I have recently been studying the convergence of a sequence of random variables. However, Let $\left\{ X_{n}\right\} _{1}^{\infty}$ be a sequence of random variables defined on $\left(\Omega,F,P\right)$ where the range of each term $X_{n}$ is the singleton set $\left\{ 1+\frac{1}{n}\right\} .$ First, I wish to be able to find $\left\{ F_{X_{n}}\right\} _{1}^{\infty}$ and $\left\{ f_{X_{n}}\right\} […]

Stirling's approximation for $ \sum _{k=1}^{n/2} \frac{n!}{k!k!(n-2k)!}a^kb^kc^{n-2k}$

Suppose $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$. I am trying to use Stirling’s approximation to obtain an asymptotic (computable) formula for $$ \sum _{k=1}^{n/2} \frac{n!}{k!k!(n-2k)!}a^kb^kc^{n-2k}$$ as $n \to \infty$. If $n$ is odd then the upper limit of the sum should actually be $(n-1)/2$. The problem is how to deal with […]

Sum of ${1 – {(1/8)} + {(1×3)}/{(8×16)} – {(1×3×5)}/{(8×16×24)} + … }$

Sum of the following series (till infinite terms): ${1 – {(1/8)} + {(1×3)}/{(8×16)} – {(1×3×5)}/{(8×16×24)} + …. }$ I tried writing the general term and then proceeding… but its not helping. Can anyone help me out? Thanks in advance!!

converging subsequence on a circle

We know that any sequence on $S^1$ must have a converging extracted subsequence, as $S^1$ is compact. Now, consider the sequence $a_n=(\cos(n),\sin(n))$. Could you find explicitly a subset of the natural numbers such that the corresponding subsequence converges? I don’t even know whether it is possible to work it out, or whether there exists a […]

Distribution and expected value of a random infinite series $\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$

Can we find the distribution and/or expected value of $$S=\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$$ where $r_n$ is a uniformly distributed random integer, $r_n \in [1,n]$ and $\text{lcm}$ is the least common multiple? Or maybe an estimate is possible? Some boundaries are easy to see. For example: $$1<S \leq \sum_{n \geq 1} \frac{1}{n^2}= \frac{\pi^2}{6} \approx 1.645$$ […]

Why do we believe that $\sum_{k=1}^{\infty} x_k=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}x_{ij}$?

I have an absolutely convergent series $\sum_{k=1}^{\infty}x_k=s$. I manage somehow to index all elements of the series along two dimensions, so each element of the series $x_{k}$ is associated with a pair $(i,j)$ and $i,j \in \mathbb N$ through a bijective mapping, so I can refer to it as $x_{i,j}$ now. It is clear that […]

geometric series and generating functions

For $x > 1$, the geometric series $\sum\limits_{i = 0}^n{x}^i$ is equal to $\frac{x^{n+1}-1}{x-1}$. By getting the limit, $$ \lim\limits_{n \rightarrow \infty} \sum\limits_{i = 0}^n{x}^i = \lim\limits_{n \rightarrow \infty} \frac{x^{n+1}-1}{x-1} $$ $$ = \infty $$ However, the generating function $\sum\limits_{n \geq 0}x^n$ is equal to $\frac{1}{1-x}$ since $$ \sum\limits_{n \geq 0}{x}^n – x\sum\limits_{n \geq 0}{x}^n= […]

Questions about finite sequences of natural numbers with distinct partial sums

I have a school assignment to do, but I have no idea, where to start. I hope you can help. Here it is: We have a finite sequence $A = (a_1, a_2,\ldots, a_n)$, length of $A$ is $n$, elements of $A$ are natural numbers. Let $S(i,j)$ be partial sum of this sequence from $a_i$ to […]