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

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 […]

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

$A$ 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 […]

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 […]

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 […]

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.

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 […]

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 […]

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