Let $x_k \in \mathbb C$ for $k \in \mathbb N \cup {0}$ and let $y_k = \frac{(x_0 + x_1 + ... + x_k)}{k+1}$. We want to prove that if $x_k$ converges to $x$ ($x \in \mathbb C$) as $k \rightarrow \infty$ then $y_k$ also converges to $x$ as $k \rightarrow \infty$.  This seems like a simple interesting exercise but having a little trouble. If $x_k \rightarrow x$ then from this I tried to make conclusions about convergence of $y_k$ since it depends of $x_k$. As $k \rightarrow \infty$ then $y_k \rightarrow \frac{(x_0 + x_1 + ... + x)}{k+1}$ but how can we conclude $\frac{(x_0 + x_1 + ... + x)}{k+1} = x$? Intuitively we also notice that all the finite terms over k+1 will go to 0 as $k \rightarrow \infty$.
Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps....
How to access object attribute given string corresponding to name of that attribute I can't seem to figure out how to get past this step. Any suggestions? Automatic image format detection in PHP I am looking for a way to take a user uploaded image that is currently put in a temporary location ex: /tmp/jkhjkh78 and create a php image from it, autodetecting the format.  Is there a more clever way to do this than a bunch of try/catching with imagefromjpeg, imagefrompng, etc? When I tried to connect my Ubuntu 13.10 to my VM Centos 6.5 using ssh appear the message "Connection refused" or "No route to host" even than I change the port, noting change.    I need to use SSH, VPN and Samba, but this IP in my VM Centos do not work.  How Can I resolve this problem? I was wondering if there is such a thing as an shapefile PRJ to PostGIS SRID lookup table?  Something that can translate the most standard shapefile PRJ definitions into the likely SRID.  When using PostGIS and pgAdminIII, if you use the postgisgui to import your shapefiles, the SRID is left as "-1".  It seems like the tool should be able to parse the Esri PRJ and determine the correct (or at least a couple of options) that are the likely SRID, rather than just leave the default.  Or does the importer have the capability to reproject on the fly if you choose another SRID?  It may seem lazy on my part, but to me it seems curious that this function hasn't already been put in place.  Does anyone know if this concept is in the works, or good reason's why it has been left out? Is it possible to allow usage for an app or program for a specific times on a Mac? I was trying to achieve this using a second account (which is the admin account) and Parental Controls, but it seemed this isn't possible to do so.  I know there are some programs which you can download that help you with this, (Cold Turkey, for example). But, it would be much easier if I could do this using only tools already available on macOS. Equation numbering problems in AMSMath environments with \pause Word for a modified version of a quote How do the comment replies (aka &quot;pinging users&quot;) work?  Who can I send an @reply to? How do I respond to a specific user when entering my comment? Will they be notified? What do people mean when they talk about &quot;pinging&quot; another user?  Related: Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$ JComboBox Selection Change Listener? I created a thick glass bottle and duplicated the interior faces to make a new object to create the liquid inside.   I went in the edit mode, selected the faces I wanted to copy and duplicated them with ShiftD. Then I pressed P to  separate the duplicated faces.   Now I try to make a duplicate of that liquid to make it flow out of the bottle, and I simply need the duplicate of one face of it. So I go to edit mode and do as I did to make the liquid from the bottle faces... but as soon as I'm back to the object mode and I want to move the duplicated faces, every thing moves with it...   I played around and tried things, and while doing so I realized that if I dupe the liquid object in object mode using ShiftD and press 0 to place it where the original is, they move together if I duplicate it and place it somewhere else so that they are separated.   Also if I pick one of the two separated objects and make it intersect with the other one, release my left mouse button to place it there, and then try to move it again, they both move together, but not at the same speed.  I really don't understand what is going on and I'm sure it's really badly explained, so what I'll do is make a screen capture of my Outliner and record the behavior of the objects and upload those to help you understand. I read this book in Middle School and I can't for the life of me remember the title, but I can remember what it was about for the most part:  Takes place in the future, the government imposes laws that outlaw people from being different in appearance and skill level. For example, if you are fast or strong or tall, they will put weights on you to weigh you down so that you'll be more slower/shorter and more like the average. Another example, if you are pretty the government issues you "ugly glasses" etc.  Basically, it's all about making sure everyone is equal and punishing the extraordinary.  One of the main characters (a man) decides to literally loose himself of his shackles and weights, breaking them to display his power (if I'm remembering correctly).   Any ideas folks? Contact from wizpert.com via stackoverflow? Well today as I was going home I got a strange email, in my private inbox (email associated with S.O.) asking me if I had solved a question on S.O. that I didn't ask. To be more precise I had given an answer.   After reading:       and several other email related issues on S.O. I decided to let you all know about this because probably I am not the only one.   So the mail subject was:     Re: Deleted facebook account - Cannot access my apps?   which refers to   And the body (masked last name of sender):      Hi Jimmy,      Just wondering if you ever got this question answered.  If not, maybe   a brief chat with a html wizpert could help: link   .      Good luck getting this resolved!      Cheers,      Michael      ------------------------- Michael W%^%^GF Founder, CEO @ Wizpert   Now I know that my email address is not visible and I read about the MD5 regarding gravatar but I don't think this is the case. I thought about some methods and sure it's possible via metadata, github, etc for someone to get my mail, but I do not dare to think that this has reached an automated level, so that some service could easily get the email of S.O. members.   I didn't get annoyed by the message, since the CEO took the time to contact me :-p but someone else could get annoyed by these kinds of messages, since they come in the important inbox of gmail.   What is more strange is that Wizpert seems to be a nice site and also is active in the Careers 2.0 if I am not mistaken.   Another thing that crossed my mind is that since Wizpert needs unanswered questions to promote their services, possibly this is an action of Wizpert trying to attract people into their services.   Well that's it, I hope I did good to let you know (maybe I did't include the correct tags). What is the location of lock screen images on Windows 10? Where, or what format, are the images used by Windows 10 when logged out or in lock-screen?  I searched the entire drive for *.jpg, *.bmp, *.png, and *.tif but the one I was looking for did not appear.  At first glance, , although not exactly a duplicate, appeared to have the path I wanted.  But when I went there, the pre-login image was there, but the lock-screen image was not.  (And contrary to the claim in that answer, the files there did have extensions.)  UPDATE: I just did a lockscreen, and got an image that IS in that directory, but earlier today, it was consistently an image of an interesting footbridge that is not in that directory. Could the Infinity Gauntlet be used to make two people fall in love with each other? I am wondering if there is anything in Marvel comic book canon or in the MCU that says that the Infinity Gauntlet has the power to make someone fall in love with the person wearing the Infinity Gauntlet, or that the wearer of the Infinity Gauntlet can use the Infinity Gauntlet to make two strangers fall in love with each other? Perhaps the combined power of the Mind Stone and the Soul Stone would make two people fall in love. For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$ I'm having trouble finishing a problem on an old national competition.  As the title states, the question says asks:  Given $a,b,c \neq 0,a+b=c$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$,  Find all integers $n$ such that $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$.  I know that if it is true for $n$, it is also true for $-n$ since $(a^n+b^n+c^n)(\frac{1}{a^n+b^n+c^n})=1$ and $(a^n+b^n+c^n)(\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n})=1$ and the statment follows for $-n$ given that we can make the substitution for $n$. Thus I found out that $n=-1$ satisfies the equality.  I also know that it fails for all even $n$. If $n$ were even, $(a^n,b^n,c^n)$ would all be positive, and Titu's Lemma asserts that $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geq \frac{9}{a^n+b^n+c^n}$. Thus $n$ must be odd.  I also observed that $(2,-1,1)$ was a solution, and from here noted that this particular solution satisfies all odd $n$.  I have been trying to prove that all other solutions satisfy all odd $n$ with no success.  Any help would be greatly appreciated. Thanks! Use laptop trackpad as graphics tablet Is it possible to export the equations from Mathematica to MATLAB?