I'm a college fresh man on my winter vacation and I'm previewing the part in my next term's Mathematical Analysis that deals with the infinite series. I have therefore learned some tricks for deciding if a series converges or not. Those tricks are quite useful, but only sometimes. When confronted with some very tricky series (perhaps in a rather simple form) they just don't work out very smoothly or even cannot work at all. So my notion is that our tricks for testing convergence are actually quite weak, and that there may well be a good many series whose convergence or divergence hasn't been revealed yet. And thus I'm asking for such an example. In addition, I want to briefly mention something interesting (my own experience): Before taking an elementary calculus course at the age of 14, I feel very strongly, by my intuition, that $\displaystyle\sum_{i=1}^{\infty}\frac1i$ must be something convergent. I "tested" it on my calculator, which told me that when $i$ turns large (in fact, no more than 50 I believe) the rate at which the sum increases has already been incredibly slow. So even without any proof I thought to myself that it would be a joke if such a thing should diverge. Of course, I was wrong, and I later learned many ways to prove I was wrong. Still, it is no surprise that someone who lacks this part of knowledge and trusts  their "mathematical intuition" will, you can almost guarantee, make the same mistake as I did. I simply get this feeling that when we are confronted with a brand new series which none of our tricks can work out (and intuition or technology is of no good help), it is just grossly difficult, or even not quite possible for us to decide its convergence or otherwise. So I firmly believe there are still some tricky ones that still bother us nowadays.
Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will eventually be shown to converge or diverge? EDIT: People were kind enough to point out that without imposing restrictions on the terms it's trivial to find such &quot;open problem&quot; sequences. So, to clarify, what I had in mind were sequences whose terms are composed of &quot;simple&quot; functions, the kind you would find in an introductory calculus text: exponential, factorial, etc.
An idempotent operator is compact if and only if it is of finite rank Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank. Can you launch Remote Desktop via URL? How do I pull from a Git repository through an HTTP proxy? Note: while the use-case described is about using submodules within a project, the same applies to a normal git clone of a repository over HTTP.  I have a project under Git control.  I'd like to add a submodule:  git submodule add http://github.com/jscruggs/metric_fu.git vendor/plugins/metric_fu   But I get  ... got 1b0313f016d98e556396c91d08127c59722762d0 got 4c42d44a9221209293e5f3eb7e662a1571b09421 got b0d6414e3ca5c2fb4b95b7712c7edbf7d2becac7 error: Unable to find abc07fcf79aebed56497e3894c6c3c06046f913a under http://github.com/jscruggs/metri... Cannot obtain needed commit abc07fcf79aebed56497e3894c6c3c06046f913a while processing commit ee576543b3a0820cc966cc10cc41e6ffb3415658. fatal: Fetch failed. Clone of 'http://github.com/jscruggs/metric_fu.git' into submodule path 'vendor/plugins/metric_fu'   I have my HTTP_PROXY set up:  c:\project&gt; echo %HTTP_PROXY% http://proxy.mycompany:80   I even have a global Git setting for the http proxy:  c:\project&gt; git config --get http.proxy http://proxy.mycompany:80   Has anybody gotten HTTP fetches to consistently work through a proxy?  What's really strange is that a few project on GitHub work fine ( for example), but others consistently fail ( for example). This question was prompted by questions about ls' -1 option and the recurring tendency of people to ask question and answers that includes processing the output of ls.   This reuse of the output ls seems understandable, e.g.: if you know how to sort a list of files with ls you might want to use the output in that way as input for something else.  If those Q&amp;A don't include a reference to the file name list produced consisting of nicely behaving file names (no special characters like spaces and newlines), they are often commented upon by someone pointing out the danger of the command sequence not working when there are files with newlines, spaces etc.  find, sort and other utilities solve the problem of communicating "difficult" file names to e.g. xargs by using an option to separate the file names with the NUL character/byte which is not a valid character in file name (the only one in addition to /?) on Unix/Linux filesystems.  I looked to the man page for ls and the output for ls --help (which has more options listed) and could not find that ls (from coreutils) has an option to specify NUL separated output. It does have a -1 option which can be interpreted as "output file names separated by newline")  Q: Is there are technical or philosophical reason why ls does not have a --zero or -0 option that would "output file names separated by NUL"?  If you do something that only outputs the file names (and not use e.g. -l) that could make sense:  ls -rt -0 | xargs -r0 …   I could be missing something why this would not work, or is there an alternative for this example that I overlooked and that is not much more complicated and/or obscure.    Addendum:  Doing ls -lrt -0 probably does not make much sense, but in the same way that find . -ls -print0 does not, so that is not a reason to not provide a -0/-z/--zero option. How many players can play split-screen, and what game modes are (or aren't) available?   How many players can play local co-op? How many players can play local multiplayer? How many players can play online multiplayer from a single console? Let $A$ be a $3\times 4$ and $b$ be a $3\times 1$ matrix with integer entries.Suppose that the system $Ax=b$ has a complex solution. Then which of the following are true? (CSIR December 2014)         $Ax=b$ has an integer solution.    $Ax=b$ has a rational solution.   The set of real solutions to $Ax=0$ has a basis consisting of rational solutions.   If $b$ is not equal to zero then $A$ has positive rank.      Is it possible to say that 1 and 2 are true as $Ax=b$ has a complex solution? If not  what we have to understand from the statement "the system $Ax=b$ has a complex solution " ? What about 3 and 4  ? Why can't a single ReLU learn a ReLU? As a follow-up to  I simplified even more and tried to train a single ReLU (with random weight) to a single ReLU.  This is the simplest network there is, and yet half the time it fails to converge.  If the initial guess is in the same orientation as the target, it learns quickly and converges to the correct weight of 1:      If the initial guess is "backwards", it gets stuck at a weight of zero and never goes through it to the region of lower loss:        I don't understand why.  Shouldn't gradient descent easily follow the loss curve to the global minima?  Example code:  from tensorflow.keras.models import Sequential from tensorflow.keras.layers import Dense, ReLU from tensorflow import keras import numpy as np import matplotlib.pyplot as plt  batch = 1000   def tests():     while True:         test = np.random.randn(batch)          # Generate ReLU test case         X = test         Y = test.copy()         Y[Y &lt; 0] = 0          yield X, Y   model = Sequential([Dense(1, input_dim=1, activation=None, use_bias=False)]) model.add(ReLU()) model.set_weights([[[-10]]])  model.compile(loss='mean_squared_error', optimizer='sgd')   class LossHistory(keras.callbacks.Callback):     def on_train_begin(self, logs={}):         self.losses = []         self.weights = []         self.n = 0         self.n += 1      def on_epoch_end(self, batch, logs={}):         self.losses.append(logs.get('loss'))         w = model.get_weights()         self.weights.append([x.flatten()[0] for x in w])         self.n += 1   history = LossHistory()  model.fit_generator(tests(), steps_per_epoch=100, epochs=20,                     callbacks=[history])  fig, (ax1, ax2) = plt.subplots(2, 1, True, num='Learning')  ax1.set_title('ReLU learning ReLU') ax1.semilogy(history.losses) ax1.set_ylabel('Loss') ax1.grid(True, which="both") ax1.margins(0, 0.05)  ax2.plot(history.weights) ax2.set_ylabel('Weight') ax2.set_xlabel('Epoch') ax2.grid(True, which="both") ax2.margins(0, 0.05)  plt.tight_layout() plt.show()     Similar things happen if I add bias: 2D loss function is smooth and simple, but if the relu starts upside down, it circles around and gets stuck (red starting points), and doesn't follow the gradient down to the minimum (like it does for blue starting points):    Similar things happen if I add output weight and bias, too.  (It will flip left-to-right, or down-to-up, but not both.) When is "self" required? I have been using classes for only a short while and when I write a method, I make all variables reference self, e.g. self.foo.  However, I'm looking through the wxPython in Action book and notice that "self" isn't used all the time. For example:   import wx  class TextFrame(wx.Frame):     def __init__(self):         wx.Frame.__init__(self, None, -1, 'Text Entry Example',             size=(300, 100))         panel = wx.Panel(self, -1)         basicLabel = wx.StaticText(panel, -1, "Basic Control:")         basicText = wx.TextCtrl(panel, -1, "I've entered some text!",             size=(175, -1))         basicText.SetInsertionPoint(0)         pwdLabel = wx.StaticText(panel, -1, "Password:")         pwdText = wx.TextCtrl(panel, -1, "password", size=(175, -1),             style=wx.TE_PASSWORD)         sizer = wx.FlexGridSizer(cols=2, hgap=6, vgap=6)         sizer.AddMany([basicLabel, basicText, pwdLabel, pwdText])         panel.SetSizer(sizer)   The one below does use "self".  import wx class ButtonFrame(wx.Frame):     def __init__(self):         wx.Frame.__init__(self, None, -1, 'Button Example',             size=(300, 100))         panel = wx.Panel(self, -1)         self.button = wx.Button(panel, -1, "Hello", pos=(50, 20))         self.Bind(wx.EVT_BUTTON, self.OnClick, self.button)         self.button.SetDefault()     def OnClick(self, event):         self.button.SetLabel("Clicked")   If I remember correctly, "self" is reference to a particular instance of the class, so when is it not necessary? Is there a general rule of thumb? Since a few months, on one of the desktop Ubuntu 14.04 systems I use, CPU has regularly been going up to 80-100% for no apparent reason. htop would usually attribute responsability to file system processes, such as nemo or ntfs. To get CPU back down I would kill these process, which in some cases would lead to loss of access to the file system and a consequent restart.  Days ago I noticed that whenever this situation happened another process was always present also using high CPU: mediascanner-service-2.0. Some research lead me to  that simply advised its removal. So I did.  The end result has been unexpectedly positive: not only are the high CPU load episodes gone, the system is altogether much faster. The speed up of the Lens is dramatic: finding an application like Calc now takes 2 seconds at most when previously it took some 20 seconds. Finding a file takes now some 5 seconds, whereas before it took in the order of 30 seconds. Heavy programmes like Eclipse are starting up faster and workspaces are more fluid. There seems to be also an overall reduction in the frequency of disk access.  What is exactly this mediascanner2.0 package? Is it part of Unity 7 or a dependency that gets installed a posteriori? Is it possible to prevent its installation?  Update: following  it is possible to track this package down to indicator-session:  $ sudo aptitude why mediascanner2.0 i   indicator-session         Recommends indicator-applet (&gt;= 0.2) | indicator-renderer i A unity8                    Provides   indicator-renderer                             i A unity8                    Recommends unity-scope-mediascanner2                      pi  unity-scope-mediascanner2 Depends    mediascanner2.0                                  Apparently, mediascanner2.0 (and the Unity 8 shell) are part of the base 14.04 system. Left to answer is the exact function of this rogue package. English version of Ubuntu with Italian keyboard I want to install Ubuntu in ENGLISH (UK) but I have an ITALIAN keyboard, which has a different layout to the English keyboard. How can I install Ubuntu in English but select an Italian layout for the keyboard. I tried, but it would only allow me to select an English keyboard. Thanks Area 51 is miscounting my proposal followers I know of one person who is following my proposal who doesn't show up on the proposal webpage.    My proposal is for quantitative finance: .  It shows 19 people following the proposal (on the side of the page), and the total listed at the top of the page is 19 followers.  Yet one user is not listed: .  This makes me wonder if other followers are also not being counted? I am using Ubuntu 16.04, with Gnome 3.18. Using extensions I have gotten rid of the dash panel and only use the Applications Menu in the upper left-hand corner of the screen. (I don't have Unity.) What I can't figure out is how do you add a new program, move software around to different categories, and create new categories?  Also, any keyboard shortcut to access the Applications Menu? What's a recommended alternative to Quickbooks pro? I've just installed Ubuntu to replace Windows. I can't seem to find a good alternative to  to run on Ubuntu. Any help would be appreciated. Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular. I'm trying to prove that statement: Prove that $\lambda = 0$ is an eigenvalue if and only if $A$ is singular.  I'm not sure if my proof is totally correct:  Suppose that $\lambda = 0$ if det(A) = $\lambda_1 \cdot \lambda_2 . . .\lambda_n = 0$ then A is singular.  If anyone could show me a better proof that would help. I have a script that does a number of different things, most of which do not require any special privileges. However, one specific section, which I have contained within a function, needs root privileges.  I don't wish to require the entire script to run as root, and I want to be able to call this function, with root privileges, from within the script. Prompting for a password if necessary isn't an issue since it is mostly interactive anyway. However, when I try to use sudo functionx, I get:  sudo: functionx: command not found   As I expected, export didn't make a difference. I'd like to be able to execute the function directly in the script rather than breaking it out and executing it as a separate script for a number of reasons.  Is there some way I can make my function "visible" to sudo without extracting it, finding the appropriate directory, and then executing it as a stand-alone script?  The function is about a page long itself and contains multiple strings, some double-quoted and some single-quoted. It is also dependent upon a menu function defined elsewhere in the main script.  I would only expect someone with sudo ANY to be able to run the function, as one of the things it does is change passwords. How to find out the default file manager in the terminal? Is there a problem with the ClearBits data dump? Currentsection TOC in scrartcl? I like to get a toc at beginning of some of my sections, like in beamer where I can use \tableofcontents[currentsection] is there a possibility in scrartcl?  Something like this:  \documentclass[10pt,a4paper]{scrartcl} \usepackage[ngerman]{babel}  \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \tableofcontents %complete  \section{First}  \section{Second} \tableofcontents %just subsection and subsubsection \subsection{SecondSub} \subsubsection{please} \subsection{this} \subsection{SecondSub2} \subsection{parts}  \section{Third} \subsection{another}  \end{document}   Looks like:  Inhaltsverzeichnis %complete  1 First  2 Second 2.1 SecondSub.......................2 2.1.1 please........................2 2.1.2 this..........................2 2.2. SecondSub2.....................3 2.2.1 parts.........................4  3 Third 3.1 another ........................5   1 First  2 Second  Inhaltsverzeichnis %just subsection and subsubsection 2.1 SecondSub.......................2  2.1.1 please........................2  2.1.2 this..........................2 2.2. SecondSub2.....................3  2.2.1 parts.........................4  2.1 SecondSub  2.1.1 please   2.1.2 this  2.2. SecondSub2  2.2.1 parts   3 Third  3.1 another Today I started having the following issues in Pokémon GO:   I can't spin pokéStops and get rewards When I catch pokémon, even the lowest CP ones, they escape every time. I've tried using razz berries, that doesn't work. Neither does using great balls.   I do have Internet. I'm not on wifi. I can send messages and browse Internet.  NOTE: I did try to modify my gps location and I was unsuccessful. I only got a pokémon or two and deleted them immediately. I do apologize for trying to hack it but I'm now fully legit. I quite enjoy paying attention to how I design my documents and presentations. I usually spend hours thinking over and designing my slides for a workshop or presentation, so that they are aesthetically pleasing and as intuitive as possible. Likewise I recently started revising my CV I figured and I wanted to make it stand out a bit more. (Just to make it clear I don't mean making a clown of a document but just better use of colors, contrast and design elements.)  I have long wondered whether or not this is something that can backfire, since most documents in academic context are extremely plain, at least in my experience. It's very common to see the default Powerpoint slides (white bg, black arial text) or .   My question is as follows: is putting time and effort into design of academic documents something that can backfire? Will I risk being prejudged with first impressions such as "well he put much effort in the presentation his documents, perhaps because the content is sub-par"?     I realize that the question might be somewhat subjective from person to person but I encourage everyone to consider it in terms of this SE blog entry: .