if  $$z_n\to 0$$, then $$\frac{1}{n}(z_1+\cdots + z_n)\to 0$$  I'm sorry for asking this one, I've seen it in may places before but I cannot think of a way to solve it, and I don't know how to search it here. Also, I've seen the solution before for real numbers.
I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following.   Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$ \lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A. $$  Is there a clever way to write the limit to make it more approachable? Thank you.
Distance between surfaces How to install and use CDI on Tomcat? How do I configure my DNS settings in Ubuntu server? Should Stack Exchange have a spam filter? (Does it already and I’m not aware?) Of course spam posts should be flagged, and if they come up in review, rejected.  Over on Mi Yodeya during the past 24 hours I’ve helped delete and reject four spam posts through the review queue, and I can only imagine that others were posted over the course of the night. To clarify, I’m referring to edits and answers that were about 99.274% words that I don’t think any decent person would use in everyday discussions, even if someone got them really angry.   Thing is, if the only thing we can do as a community is delete his posts/reject his edits, and as moderators to ban his account, can’t he just create a new account? Further, some of these posts . Maybe the mods can ban his IP address? I’m not one, so I have no idea what kind of power they wield in that respect. But he can always just get a new computer.   I don’t have the numbers to see what this is like on other sites, but I know this doesn’t happen often on Mi Yodeya. I wonder, then, whether this is the optimal solution, or whether an automatic spam filter be put into place - say, a new review queue that automatically deletes posts or rejects edits that can then be voted on whether to be reposted if it’s a false alarm. Perhaps it can be automatically triggered by certain keywords (like ****, %$#^, or @$+^).   Now, all of these posts that I voted to delete/reject came up in Suggested Edits and First Posts. Maybe such edits would show up in Low Quality in general, and that just wasn’t the case here because of precedence rules. I don’t know. But perhaps the above would be a better solution, if this is even as big of a problem as I’m making it out to be. Does there exist a symmetric matrix $A$  such that $2^{\sqrt{n}}\le |\operatorname{Tr}(A^n)|\le2020 \cdot2^{\sqrt{n}}$ for all $n$? I think no. The trace of $A^n$ equals $\sum\limits_{i=1}^n\lambda_i^n$ where $\lambda_i$ are the eigenvalues of $A$. Now, if the absolute value of trace of $A$ is bounded below by $2$, then I think the trace of $A^n$ will grow infinitely. Am I right? Thanks beforehand. Graph of a continuous function is closed Irreducible cyclotomic polynomial I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$? I got a problem while making a model of a product. I have tried a lot even seen lots of tutorial but did not get the thing that i want. I am using Blender 9.92 and I am just a beginner I got my hands on Blender 7-8 days ago. I've just basic knowledge about I am trying to model a product &quot;Mattress Topper&quot;   This is the exact look of the product, Help me how i can model this product and render realistic images. If I select a feature and run self.iface.actionIdentify().trigger() it will open the identify window but won't open the selected feature's data, how can I make it open the data of the selected feature? Get dateformat based on Locale Proof of continuity of Thomae Function at irrationals. In :  $$ \begin{align} t(x) = \begin{cases} 0 &amp; \text{if $x$ is irrational}\\ \frac{1}{n} &amp; \text{if $x = \frac{m}{n}$ where $\gcd(m,n) = 1$} \end{cases} \end{align} $$  I can prove the discontinuity at rational $b$ by taking a sequence of irrationals $x_n$ which converge to $b$.   But while going through an argument for continuity at irrationals.  I found this in a book.      On the other hand if $b$ is an irrational number and $\epsilon &gt; 0$   then there is a     natural number $n_0$ such that $1/n_0 &lt; \epsilon$.   There are only finite number of rationals with denominator less than   $n_0$ in the interval $(b-1,b+1)$. Hence we can find a $\delta &gt; 0$   such that $\delta$ neighbourhood of $b$ contains no rational with   denominator less than $n_0$.   I understand the rest of the proof. But I am unable to prove the emphasized text. Although I find it intuitive. What is the difference between old style and new style classes in Python?  When should I use one or the other? How do airlines determine ticket prices? "Be yet to do" vs "have yet to do" This theory has yet to be proven. This theory is yet to be proven. This is yet to be done. This has yet to be done. I have yet to spend summer in the mountains. I am yet to spend summer in the mountains.  Are they all correct? How are &quot;have yet to do&quot; and &quot;be yet to do&quot; different? Thank you! How do you rotate a two dimensional array? Inspired by , say you have a 4x4 two dimensional array, write a function that rotates it 90 degrees. Raymond links to a solution in pseudo code, but I'd like to see some real world stuff.  [1][2][3][4] [5][6][7][8] [9][0][1][2] [3][4][5][6]   Becomes:  [3][9][5][1] [4][0][6][2] [5][1][7][3] [6][2][8][4]   Update: Nick's answer is the most straightforward, but is there a way to do it better than n^2? What if the matrix was 10000x10000? I just bought a new HP Pavilion dv7t Quad Edition laptop that has HDMI out. I already have a couple of monitors that are VGA and I'd like to hook up one of these to dual monitors.  I've read some sketchy reviews of products that try to convert directly from HDMI (digital) to  VGA (analog), so I was wondering if someone has a good recommendation on how to hook these two together. How to evaluate  $$\det\begin{bmatrix}1+x_1y_1&amp;\cdots &amp; 1+x_1y_n\\ \vdots &amp; \ddots &amp; \vdots \\ 1+x_ny_1 &amp; \cdots &amp; 1+x_ny_n\end{bmatrix}$$?  I tried factorisation but I have no idea. I could not factor anything out... I need a hint to kick off.   Edit: A conjecture due to multilinearity: if we let $v_j=(x_1y_j,\cdots,x_ny_j)^T$, then my guess would be $$\det\begin{bmatrix}1+x_1y_1&amp;\cdots &amp; 1+x_1y_n\\ \vdots &amp; \ddots &amp; \vdots \\ 1+x_ny_1 &amp; \cdots &amp; 1+x_ny_n\end{bmatrix}=\det(1,v_2,\cdots,v_n)+\det(v_1,1,\cdots,v_n)+\cdots+\det(v_1,v_2,\cdots,1)=\color{red}{0}$$ where $1$ denote $(1,\cdots,1)^T\in \Bbb K^n$,$\Bbb K$ the underlying number field. Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$ As you know, we can enable the spell checker on single line text boxes in the following way:    We have to enable it every time for checking words.  How can we enable it forever, so it doesn't need to be enabled each time? Is it possible to shutdown remote Computers without inputing the password? I used the following bash to shutdown remote computers,but for each computer,I must input the password for it.Is it possible to input the password Automatically?   #!/bin/bash  ssh -t root@192.168.1.11 "sudo poweroff" ssh -t root@192.168.1.12 "sudo poweroff" ssh -t root@192.168.1.13 "sudo poweroff" ssh -t root@192.168.1.14 "sudo poweroff" ssh -t root@192.168.1.15 "sudo poweroff" ssh -t root@192.168.1.16 "sudo poweroff" ssh -t root@192.168.1.17 "sudo poweroff" ssh -t root@192.168.1.18 "sudo poweroff" ssh -t root@192.168.1.19 "sudo poweroff" ssh -t root@192.168.1.20 "sudo poweroff" ssh -t root@192.168.1.21 "sudo poweroff" ssh -t root@192.168.1.22 "sudo poweroff" ssh -t root@192.168.1.23 "sudo poweroff" ssh -t root@192.168.1.24 "sudo poweroff" ssh -t root@192.168.1.25 "sudo poweroff" ssh -t root@192.168.1.26 "sudo poweroff"