I want to prove the following statement:  Suppose that $\displaystyle\sum_{k=1}^{\infty}a_k$ converges, where $(a_k)_{k\in\mathbb{N}}\subseteq\mathbb{R}$ is monotone. Then $\displaystyle\lim_{k\to\infty}ka_k=0$.  I believe we have several cases.  For example, if $(a_k)_{k\in\mathbb{N}}$ is monotone increasing and there exists $k$ such that $a_k&gt;0$, then obviously $\displaystyle\sum_{k=1}^{\infty}a_k$ is not convergent.   Then, we could conclude that if some $a_k&gt;0$ then we can suppose that $(a_k)_{k\in\mathbb{N}}$ is monotone decreasing. By the same argument, we can conclude that if some $a_k&lt;0$ then $(a_k)_{k\in\mathbb{N}}$ must be monotone increasing.  So, I believe we only need to take care of the case where $a_k\ge 0$ for each $k\in\mathbb{N}$ and $(a_k)_{k\in\mathbb{N}}$ is monotone decreasing (the other case would be symmetric). Any hint to prove this? I have been thinking a lot ot time...  Thanks.
I need a little hint in a proof on convergence of a sequence. The problem I have is: Suppose $(x_n)$ is a monotone decreasing sequence of real numbers such that $\sum x_n$ converges, prove that:  $$\lim_{n \to \infty} nx_n=0$$  My idea is: proving this fact aims to prove that given $\epsilon &gt;0$ there's some natural $n_0 \in \mathbb{N}$ such that if $n &gt;n_0$ we have $|nx_n|&lt;\epsilon$.  Now, I know that $\sum x_n$ converges, so that given $\epsilon'&gt;0$ there's some $k_0 \in \mathbb{N}$ such that if $k &gt; k_0$ we have:  $$\left|\sum_{i=1}^{k}x_i - S\right|&lt;\epsilon'$$  Where $S = \sum x_n$. Now, since $(x_n)$ is monotone decreasing we know that we must have $x_1 &gt; \cdots &gt; x_k$ so that the sum of all $x_i$ should be less or equal to $k x_k$. So I know that I have:  $$\left|\sum_{i=1}^{k}x_i - S\right|\leq\left|\sum_{i=1}^{k}x_i\right|+|S|\leq|kx_k|+|S|$$  I feel that the proof will come from this, however I'm stuck at this point. Can someone give just a little hint on how to proceed from here?  Thanks very much in advance for your help.
how to set camera to the same view as I turn my cursor? I want to turn the camera to get the view I want, but everytime I turn the camera, I need to go to render window to check if it's the view I want. I turn the view with my mouse cursor, but that does not affect the actual camera view in rendering, can I set the camera view the same as I turn my cursor? latex gives me the error: Ouch---my internal constants have been clobbered As the title states, compiling with latex produces the error, right away:  Ouch---my internal constants have been clobbered!---case 14   I have no idea what this means... I use TexLive 2012. I updated my texmf.cnf and ran fmtutil-sys and texhash. I guess texhash may have messed it up?  Any ideas?  As requested a minimal working example:  \documentclass{article} \begin{document}     Any tex file. \end{document} How to install the previous version of a .deb package and pin it? I am trying to install the latest version of subversion on Sid, and because it has a bug I receive a warning and I abort the installation. How do I get to locate the previous version version, install it and pin until the bug is resolved?  root@server01:~# apt-get install subversion   Reading package lists... Done                                                                                                                                                                                  Building dependency tree                                                                                                                                                                                       Reading state information... Done                                                                                                                                                                              Suggested packages:                                                                                                                                                                                              db5.3-util subversion-tools                                                                                                                                                                                  The following NEW packages will be installed:                                                                                                                                                                    subversion                                                                                                                                                                                                   0 upgraded, 1 newly installed, 0 to remove and 205 not upgraded.                                                                                                                                               Need to get 0 B/981 kB of archives.                                                                                                                                                                            After this operation, 4,844 kB of additional disk space will be used.                                                                                                                                          Retrieving bug reports... Done                                                                                                                                                                                 Parsing Found/Fixed information... Done                                                                                                                                                                        critical bugs of subversion (-&gt; 1.9.2-2) &lt;Outstanding&gt;                                                                                                                                                          b1 - #803725 - subversion: dump-load of a repository modifies verbose log output: M line lost serious bugs of subversion (-&gt; 1.9.2-2) &lt;Outstanding&gt;  b2 - #803589 - FTBFS with ruby2.2 (only) Summary:  subversion(2 bugs) Are you sure you want to install/upgrade the above packages? [Y/n/?/...] n ********************************************************************** ****** Exiting with an error in order to stop the installation. ****** ********************************************************************** E: Sub-process /usr/sbin/apt-listbugs apt returned an error code (10) E: Failure running script /usr/sbin/apt-listbugs apt I have an UTF-8, polish text in .tex file. When I open it with my LEd (LaTeX editor), it shows ok (all characters can be seen). But when I try to compile it, I get error:  Package inputenc Error: Keyboard character used is undefined (inputenc) in inputencoding `utf8'.   I have to copy the text, cut it and paste to Notepad++, change (not convert!) encoding from UTF8 to ANSI (the polish characters will be missed and replaces with some random characters). Than copy it again and paste to LEd (in that editor, characters display bad also). But now I can compile my pdf and in it, the encoding is right again.  Is there a way to set encoding in the way I will see characters in LEd propertly and I will be able to compile it and I will get good characters in pdf?  The begging of my code:  \documentclass[12pt]{report} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{graphicx} \usepackage{amsmath,amssymb,amsfonts} \usepackage{txfonts} Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem) The black book knowledge I found a couple useful threads on this forum with respect to the issues posed by (displaying) images in email signatures-- , for example-- and I've searched all over the web, but still have not found a good solution that sufficiently addresses the problem. A client of mine would simply like his company's logo included in the signature of his emails, and the issues I've encountered can be summarized as follows:   I can export a rasterized version of the logo from AI at actual size, and it will look sharp on the desktop but pixelated/blurry on hi-density (e.g. "retina") displays such as the iPhone. As suggested in the thread I referenced, I can export the logo at 2-3 times the actual, displayed size to target the hi-density displays, but the logo will then look soft on non-hi-density displays when it is scaled down. This is a particular problem in this instance as the logo contains text, which looks terrible when juxtaposed with actual text in the browser/email client. I've considered .svg as an option, but apparently the support is not great; and in this case I'm assuming the vast majority of the users reading this client's email will be using Outlook, so something that only renders properly in iOS/webkit/etc is not a viable option.   I'm at a loss at this point and wondering if there is any other possible option out there. I'm not sure, for instance, if it's possible to implement a hi-density image with lo-res fallback in an email signature?  Any suggestions/insights here are much appreciated. It's bordering on comical how difficult this task has turned out to be. Take the following class as an example:  class Sometype {     int someValue;      public Sometype(int someValue)     {         this.someValue = someValue;     } }   I then want to create an instance of this type using reflection:  Type t = typeof(Sometype); object o = Activator.CreateInstance(t);   Normally this will work, however because SomeType has not defined a parameterless constructor, the call to Activator.CreateInstance will throw an exception of type MissingMethodException with the message "No parameterless constructor defined for this object." Is there an alternative way to still create an instance of this type? It'd be kinda sucky to add parameterless constructors to all my classes. Well, you can see it for yourself. The badge multiplier is a bit misaligned. Can you fix this please? (I know it's kinda a minor issue but it's bugging me and it's somehow off)    If it helps, I'm using Chrome version 44.0.2403.125 m. (Also repro'ed on FireFox 39.0 and )    For no good reason, that's a screenie from 's profile page. Is there a way to change the process group of a running process? Is there a way to change PID, PPID, SID of a running process? It would make sense for the answer to be no, but I'd like to make sure. CAST and IsNumeric Why would the following query return "Error converting data type varchar to bigint"?  Doesn't IsNumeric make the CAST safe?  I've tried every numeric datatype in the cast and get the same "Error converting..." error.  I don't believe the size of the resulting number is a problem because overflow is a different error.  The interesting thing is, in management studio, the results actually show up in the results pane for a split second before the error comes back.  SELECT CAST(myVarcharColumn AS bigint)   FROM myTable   WHERE IsNumeric(myVarcharColumn) = 1 AND myVarcharColumn IS NOT NULL   GROUP BY myVarcharColumn   Any thoughts? If I put a DX lens on a Nikon D700, it will reduce megapixels to 5.1.  I understand that FX pixels are about 2.5 times larger than DX pixels. Would I be correct to assume that, therefore, my FX image using 5.1 megapixels will produce quality equivalent to at least a 10 megapixel image on a DX camera, all other things being equal? How can I get Google Sheets to treat my form-submitted strings as plain text and preserve leading zeros? I have Google Forms where users enter information, some of which contains leading zeros (e.g. zip codes, ID numbers).   However, when this data is saved to the Google Sheet, it doesn't treat it as plain text and strips the zero. The form fields are text fields with no validation.   I have tried to format the range of cells containing these values as plain text, but it seems Sheets does not set the formatting as plain text unless there is already text in a cell. So I can format a column as plain text, submit the form, and the leading zero will still be stripped.   I tried writing an apps script to deal with this but there's no good way for me to deploy it to the users who would need it.   How can I get around this without having to instruct users to prepend a single quote to their form entry? This is such a small thing, but I can't find a good simple solution to what it seems must be a common problem. If I can't get this to work I may have to drop Google Forms/Sheets altogether. Microsoft Exchange 2013 is not receiving internet email How do I efficiently get "the most recent corresponding row"? I have a query pattern that must be very common, but I don't know how to write an efficient query for it.  I want to look up the rows of a table that correspond to "the most recent date not after" the rows of another table.  I have a table, inventory say, which represents the inventory I hold on a certain day.  date       | good | quantity ------------------------------ 2013-08-09 | egg  | 5 2013-08-09 | pear | 7 2013-08-02 | egg  | 1 2013-08-02 | pear | 2   and a table, "price" say, which holds the price of a good on a given day  date       | good | price -------------------------- 2013-08-07 | egg  | 120 2013-08-06 | pear | 200 2013-08-01 | egg  | 110 2013-07-30 | pear | 220   How can I efficiently get the "most recent" price for each row of the inventory table, i.e.  date       | pricing date | good | quantity | price ---------------------------------------------------- 2013-08-09 | 2013-08-07   | egg  | 5        | 120 2013-08-09 | 2013-08-06   | pear | 7        | 200 2013-08-02 | 2013-08-01   | egg  | 1        | 110 2013-08-02 | 2013-07-30   | pear | 2        | 220   I know one way of doing this:  select inventory.date, max(price.date) as pricing_date, good from inventory, price where inventory.date &gt;= price.date and inventory.good = price.good group by inventory.date, good   and then join this query again to inventory. For large tables even doing the first query (without joining again to inventory) is very slow.  However, the same problem is quickly solved if I simply use my programming language to issue one max(price.date) ... where price.date &lt;= date_of_interest ... order by price.date desc limit 1 query for each date_of_interest from the inventory table, so I know there is no computational impediment.  I would, however, prefer to solve the whole problem with a single SQL query, because it would allow me to do further SQL processing on the result of the query.  Is there a standard way to do this efficiently? It feels like it must come up often and that there should be a way to write a fast query for it.  I'm using Postgres, but an SQL-generic answer would be appreciated. I've been working on the Android SDK platform, and it is a little unclear how to save an application's state. So given this minor re-tooling of the 'Hello, Android' example:  package com.android.hello;  import android.app.Activity; import android.os.Bundle; import android.widget.TextView;  public class HelloAndroid extends Activity {    private TextView mTextView = null;    /** Called when the activity is first created. */   @Override   public void onCreate(Bundle savedInstanceState) {     super.onCreate(savedInstanceState);      mTextView = new TextView(this);      if (savedInstanceState == null) {        mTextView.setText("Welcome to HelloAndroid!");     } else {        mTextView.setText("Welcome back.");     }      setContentView(mTextView);   } }   I thought it would be enough for the simplest case, but it always responds with the first message, no matter how I navigate away from the app.  I'm sure the solution is as simple as overriding onPause or something like that, but I've been poking away in the documentation for 30 minutes or so and haven't found anything obvious. I would like to port Ubuntu Touch to a device that is currently not supported. What are the requirements for doing this and where do I begin? For usability reasons, for example in , the 'viewed' counter should show as  ...instead of... . It's much harder to decipher that there are a million views on the question just by looking at it.  Some people in the comments pointed out internationalization concerns with it. I do think that since there might be a problem with it, that spaces would probably be a better idea. I moved postgresql's data directory by following the following steps:   Stop postgres cp -a source_data_directory destination_data_directory export PGDATA=destination_data_directory Changing data directory to destination_data_directory within postgresql.conf pg_ctl start   But when I run: "pg_ctl start" I get the following error. Can someone please help me figure out the error.   server starting postgres@Server:/usr/lib/postgresql/9.1/bin$ postgres cannot access the server configuration file "/home/destination_data_directory/postgresql.conf": No such file or directory Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$