I have been told, that Cauchy second limit theorem doesn't always work both ways - if the limit of $n$th root of sequence exist it doesn't immediately mean the limit of $a_{n+1}/a_n$ is equal to nth root limit. Can you explain why? Show an example? Thank you a lot.
Let consider the power series $$ \sum_{k=1}^\infty a_kx^k. $$ We know that it converges when  $$\lim_{n\to \infty }\sqrt[n]{|a_n|}|x|&lt; 1\quad\text{or}\quad |x|&lt; \frac{1}{\lim_{n\to \infty }\sqrt[n]{|a_n|}}$$  But D'Alembert test tells us that it converges if $$|x|&lt; \frac{1}{\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|},$$ so since the radius of convergence is unique, we should have that $$\lim_{n\to \infty }\sqrt[n]{|a_n|}=\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|.$$  I tried to prove it, but this result looks strange to me. So if it doesn't work always, how can we have those to limit as radius of convergence? I guess the equality of those two limit should be true most of the time.  So under what conditions is it true?
I hear the word architect used as a verb in the technical field and now more often in other industries and groups, for example:     We need to architect a better solution to the problem.   I am interested if this is considered acceptable usage, as I see this word being used more in published technical documents (not necessarily books, but corporate publications).  I can see how a conductor can conduct, a typist can type, but can a scientist "science"?  A scientist might research, analyze, and so forth, as an architect might design, create, build, etc.  Merriam-Webster , however, I have been told that other dictionaries exist that define architect as both a noun and a verb&nbsp;— is there a credible reference or source that authoritatively answers this question? proof that $f'$ is continuous at $a$ Let $f:I \rightarrow \mathbb{R}$ derivable in set I. if $\lim x_n = \lim y_n =a \in I$ with $x_n \neq y_n$ and $\lim \frac{f(y_n)-f(x_n)}{y_n - x_n} = f'(a)$ proof that $f'$ is continuous at $a$  my attempt $x_n \neq y_n$ so by MVT $\exists z_n \in I$ like $x_n \geq z_n \geq y_n$ with $\frac{f(y_n)-f(x_n)}{y_n - x_n}=f'(z_n)$ for each $x_n$ and  $y_n$ then $f'(a) = \lim\frac{f(y_n)-f(x_n)}{y_n - x_n}= \lim f'(z_n)$   Using $\lim x_n = \lim y_n = a$ and $x_n \geq z_n \geq y_n \Rightarrow z_n \rightarrow a$  if $ \lim_{x\rightarrow a} f'(x) \neq f'(a)$ then $\exists \epsilon &gt; 0$ such $\forall n\in \mathbb{N}$, $z_n \in [x_n,y_n]$ $0 &lt; |z_n - a| &lt; \frac{1}{n}$ and $|f'(z_n) - f'(a)| \geq \epsilon$. So $z_n \rightarrow a$ but will do not have $\lim f'(z_n) = f'(a)$  so $\lim f'(z_n) = f'(a) = \lim_{x\rightarrow a} f'(x)$ when $z_n \rightarrow a$  and thats implies $f'$ continuous at $a$  my question is, that's correct ? if not, can you help me? thanks Why is my "Committed" memory so much higher than my actual RAM space? I have run two virus scans and neither of them have found anything so far (Malware Bytes and Avira Rescue). During gaming I will get Windows popups saying that my system is low of memory and recommends closing the game. This can also happen while browsing. My computer seems to become sluggish a lot and will hang up often, such as when clicking a link from a messenger program to open in my browser.   I am unsure what is causing this. My memory usage on the processes tab doesn't add up to such a high percentage of my actual memory.    As a side note, is my Disk Usage supposed to spike up to 100% sometimes? Research seems to say so. (p.236)  has a proof that the equation $e^z-z=0$ has infinitely many complex solutions:    I'm curious if there are any particularly elegant ways to see this, other than that given in the text. An example of grey and white note that I mean is, like when you copy something, a small grey background and white lettered bubble appears for a few moments saying 'copied to clipboard'. My phone has some sort of malfunction where it dispays that prompt for several minutes at a time. Very annoying. How do I disable that? I am using a Samsung Galaxy S9. How can I keep track of a decimal number that indicates the current paragraph's place in a logical hierarchy and print it in the margin of each paragraph?  (To yield a numbering in the style of Tractatus Logico-Philosophicus).  \def\on{} \def\down{} \def\up{}  \on This is the first paragraph, paragraph number 1.  \on This is paragraph number 2.  \down This is paragraph number 2.1  \down 2 This is 2.101  \up This is 2.11  \down 2.111  \up 3 3  \bye   If \down is starred, begin with zero.  If \up is followed by an integer, it should take it as an argument, unless there is a newline between \up and the integer.  An indefinite number of decimals should be possible.  PS.  I apologize for changing the question after (2) answers have been given, but I realize I failed to appreciate a feature of Wittgenstein's numbering: the last decimal is never 0.  Therefore, \down &lt;n&gt; is better than \down*, as now demonstrated above.  I think the 0 is supposed to indicate an additional level of unimportance. Add a "back to top" link The left-nav is great - I like it, users who don't can disable it. Perfect world. Now it's stickied it's always there too, which is really useful.  When I'm navigating in areas not directly linked to in the left nav and have scrolled down, I often want to re-use the other navigation. This means scrolling up. Which is fine, of course it's how websites work.  However, if the left-nav had a "Top" link which took the user to the very top of the page it would be great for getting back to specific contextual menus and things that are not linked to in the left-nav.  For example, if I'm viewing questions on a specific tag and I've scrolled down, I sometimes want to get back up to the top to remove a tag, or change from "Latest" to "Featured" etc. Although clicking the "Questions" link in the left-nav takes me back to the top, it removes the tag(s) I was searching with.  There are other use cases where this would be useful, like long questions and answers (Etc). So I think a "Top" link in the left nav would be very useful.    I think this is different to the  as it doesn't mention the left-nav, and my question is suggesting this idea because we have a good place to put it and it'll be a useful addition to the left-nav. How to build a dmg Mac OS X file (on a non-Mac platform)? Is it possible to build a .dmg file (for distributing apps) from a non-Mac platform? And if yes, how? How to handle a native OS (Ubuntu) file upload window in selenium webdriver with python? This is my first time here but I have already posted this question on SO. I keep getting solutions for Windows + Java and not for Linux or python for that matter.  I am using Selenium and Python to automate a file upload application for the past many hours but nothing helpful!   The HTML DOM structure for this file upload is an input tag with its type attribute value = 'file'. I know I can make use of the send_keys.   The problem is I am not supposed to use send_keys() (as per the question I am trying to solve). The assumption is there are other ways of doing it.  Can anyone suggest me alternative ways to upload the file? I do get solutions for windows but not for Linux(Ubuntu). I know there are applications/tools to automate GUI components like AutoIt but there is nothing for Linux close to AutoIt.  I can say I have done my homework to find a solution. Only after not finding anything useful, I am asking this question, please help! What is the difference between these two ways to calculate average velocity? Average velocity:  $$v_{\rm avg,1}=\frac{v_{\rm final}+v_{\rm initial}}{2}$$   and average velocity:   $$v_{\rm avg,2} =\frac{\rm total\;displacement}{\rm time \;taken}=\frac{\Delta x}{\Delta t} $$  What is the difference between them and when do we use them? Is there any way to tell Google to use a particular account for all Google services by default? I typically use 2 Google accounts simultaneously in my browser. So whenever I fire up any Google service like Google Maps or YouTube or Google Search, it uses one of the accounts. The problem is that I don't want that account to be used and no matter how many times I switch the account for these services, they seem to default to same account.  Is there any way to tell Google to use a particular account for all Google services by default?    I am not talking about using Google account to sign in to third party websites. I am talking only about Google's own services.  I am using Chrome if anyone is going to suggest client-side solution. Is GTA Online cross-platform? Grand Theft Auto Online exists on five different platforms: XBox 360, PS3, XBox One, PS4, and PC. Do any of these play on the same servers, allowing people on differing platforms to play together? How to prove that if A, B, C are submodules of M module then $$ A \cap (B+C) = (A\cap B)+C \iff C \subseteq A $$  Any help would be appreciated! How to reinstall kernels after accidental deletion with a separate boot partition and an encrypted LVM volume? Difference between static class and singleton pattern? What real (i.e. practical) difference exists between a static class and a singleton pattern?  Both can be invoked without instantiation, both provide only one "Instance" and neither of them is thread-safe. Is there any other difference? How to concatenate items in a list to a single string? Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it. EntityFieldQuery order by title Why 220V and 110V? Why did the industry choose these seemingly unintuitive values of 110V and 220V for the mains AC? Is there a physical reason why they aren't simply 100V and 200V? I have a lidar file (file.las) with the following spatial reference:  Spatial Reference:            PROJCS["NAD83 / UTM zone 11N",     GEOGCS["NAD83",         DATUM["North_American_Datum_1983",             SPHEROID["GRS 1980",6378137,298.2572221010002,                 AUTHORITY["EPSG","7019"]],             AUTHORITY["EPSG","6269"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4269"]],     PROJECTION["Transverse_Mercator"],     PARAMETER["latitude_of_origin",0],     PARAMETER["central_meridian",-117],     PARAMETER["scale_factor",0.9996],     PARAMETER["false_easting",500000],     PARAMETER["false_northing",0],     UNIT["metre",1,         AUTHORITY["EPSG","9001"]],     AUTHORITY["EPSG","26911"]]   Then I reproject it to WGS84 with las2las:  las2las --a_srs EPSG:26911 --t_srs EPSG:4326 -i file1.las -o output.las   And I get:   Spatial Reference:                GEOGCS["WGS 84",         DATUM["WGS_1984",             SPHEROID["WGS 84",6378137,298.257223563,                 AUTHORITY["EPSG","7030"]],             AUTHORITY["EPSG","6326"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4326"]]   But when I try to revert the transformation: from WGS84 to NAD83/UTM 11N, I get an error:  las2las --a_srs EPSG:4326 --t_srs EPSG:26911 -i output.las -o wgs2utm.las ERROR 1: latitude or longitude exceeded limits error: Could not project point for ReprojectionTransform::latitude or longitude exceeded limits0.   Why I got this error, if I am just reversing the transformation (originally from NAD83 to WGS84, and then from WGS84 to NAD83)?.