Articles of sequences and series

Cauchyness vs. Double Limits

Maybe there are some textbooks which might treat cauchyness by taking double limits… My question: Is it sufficient and necessary to consider the double limit: $$x_n\quad \text{cauchy}\quad \Leftrightarrow \quad \lim_{(m,n)}d(x_m,x_n)=0$$ Moreover, if so, does it suffice as well to consider the limits successively and interchanged: $$\lim_m\lim_n d(x_m,x_n)=0 \quad\text{and}\quad \lim_n\lim_m d(x_m,x_n)=0$$ Thanks in advance. Cheers, Alex. […]

Find partial sums of the series $12+105+1008+10011+\dots$

Find the sum of $n$ terms of this series- $$12+105+1008+10011+…..$$ I did not understand that how should I proceed with this problem.

Pick the highest of two (or $n$) independent uniformly distributed random numbers – average value?

With “random number” I mean an independent uniformly distributed random number. One Picking one random number is easy: When I pick a random number from $x$ to $y$, the average value is $(x+y)/2$. Two I’m no maths expert, nevertheless I was able to work out a solution for whole numbers: When I pick the highest […]

An intuitive reasoning for 1+2+3+4+5… + ∞ = -1/12?

This question already has an answer here: Why does $1+2+3+\cdots = -\frac{1}{12}$? 16 answers How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate] 4 answers

Largest binary sequence with no more than two repeated subsequences

$A$ is an ordered sequence of elements $a_i = 0, 1$ containing no more than two adjacent repeated subsequences $[a_i, a_{i + k})$. What is the longest sequence $A$? Is it even finite? For example, the subsequence $\{0\}$ is found three times in a row in $\{0, 0, 0\}$, and $\{1, 0\}$ is found in […]

A sequence of functions $\{f_n(x)\}_{n=1}^{\infty} \subseteq C$ that is pointwise bounded but not uniformly bounded.

We were talking about pointwise bounded vs. uniformly bounded in my analysis class, and this question came up. The problem is that we are working on a compact set, it would be much easier if the interval was $(0, 1]$. My idea was to create a sequence of functions such that $f_n(\frac{1}{n}) = n$ and […]

Positive lower bound for a function defined as a series?

How we could know that for the function $f(x)$ below if there exist (not exist) a constant $C>0$ such that $f(x)\geq C$ for all $x\in \mathbb R$, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x-n)^{2}+1}$$ (i.e., $\lim_{x\to\pm\infty}f(x)\neq 0$) EDIT: I got something which I’m not sure if its correct or not: If $x>0$ is too large then there is $n_{o}$ such that […]

Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can’t seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.

Find two constant$c_1, c_2$ to make the inequality hold true for all N

Show that the following inequality holds for all integers $N\geq 1$ $\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$ where $c_1,c_2$ are some constants. It’s obvious that by dividing $\sqrt{N}$, we can get $\left|\frac{1}{N}\sum_{n=1}^N \frac{1}{\sqrt{n/N}} -2 – \frac{c_1}{\sqrt{N}}\right|\leq \frac{c_2}{N}$ the first 2 terms in LHS is a Riemann Sum of $\int_{0}^1\frac{1}{\sqrt{x}}dx$. But what is the next step? I have not seen a […]

Convergence of $\sum_{n=1}^{\infty} a^{-\frac{1}{c+1}}\prod_{k=1}^{n} \frac{a}{a+k^{c}}$ as $a\to\infty$

It looks to me, by doing numerical simulations, that $$ f(c) = \lim_{a\rightarrow \infty}a^{-\frac{1}{c+1}}\sum_{n=1}^{\infty} \prod_{k=1}^{n} \frac{a}{a+k^{c}} $$ converges and that $1\leq f(c) <2$, which was quite surprizing to me. It is easy to see that $f(0)=1$ (using geometric series) and it has been shown that $f(1)= \sqrt{\frac{\pi}{2}}$, as shown by the answer to Convergence of […]