Prove or give a counter example $ \sum_{n=1}^{\infty } a_{2n} \ and \sum_{n=1}^{\infty } a_{2n-1} $ converge than $ \sum_{n=1}^{\infty } a_{n} $
Prove that if $\sum_{n=1}^\infty a_{2n}$ and $\sum_{n=1}^\infty a_{2n-1}$ converge then $\sum_{n=1}^\infty a_n$ converges.
Adding more than 15 digits in Excel I want to add more than 20 digits in an Excel cell.  The current format of the cell is general, it converts the number to an exponential format. I tried with a number format and accounting, but when I enter more than 15 digits it gets converted to 0's.  What are the recommended steps for stopping Excel from converting data to exponential Format for 20 digits when in the general format?     Example: 12345678901234567890   Excel converts it to 1.23457E+19 in general format.  Without using ' before the value, is there any other way to keep value same? Down vote changed to up vote caused reputation meter to pass daily maximum I just experienced a minor visual bug.  I had an answer that earned 2 down votes and 22+ upvotes, earning me the daily maximum of +200. Upon making some edits to the answer at the behest of the comments, one of the users who down voted changed their vote. Now my reputation meter shows that I gained +202 for the day.    This correctly did NOT affect my reputation, as I still only gained 200, so it is merely a minor display bug.  This appears in the achievements drop down and my reputation tab on my profile. How to remove accents / diacritic marks from a string in Qt? zero matrix to the power of 0 Why $0^0=I$?   I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem  Thanks in advance. How to get domain name from URL Is verifying ISOs downloaded from the official website worthwhile? I downloaded the ISO from , selecting the default "Ubuntu Desktop" option.  The website links the page  which gives instructions how to verify ubuntu.  This seems pretty tedious, and I am wondering how realistic it is that there is a problem with ISO downloaded from the official website. I note that the process of verification itself requires me to download software that is new to me, thus introducing another attack vector on me even as I am closing another one.  For what it's worth, I am planning to use Live USB only and not to fully install Ubuntu. Does that make a difference? Where's the Strangedupe blog post? Previously, in order to explain the benefits of (certain limited kinds of) duplication on Stack Exchange sites, I've seen a blog.stackoverflow blogpost with the words "Dr. Strangedupe" in the title.  The post comes up in a search on the new blog, but       Can we please bring this post back? I'd love to link to it when , to reassure the questioner that there is value in not deleting their question because of search terms, but the blog post explained it a lot better than I could in a comment. Visual Studio C# statement collapsing When editing really long code blocks (which should definitely be refactored anyway, but that's beyond the scope of this question), I often long for the ability to collapse statement blocks like one can collapse function blocks.  That is to say, it would be great if the minus icon appeared on the code outline for everything enclosed in braces.  It seems to appear for functions, classes, regions, namespaces, usings, but not for conditional or iterative blocks.  It would be fantastic if I could collapse things like ifs, switches, foreaches, that kind of thing!  Googling into that a bit, I discovered that apparently C++ outlining in VS allows this but C# outlining in VS does not.  I don't really get why.  Even notepad++ will so these collapses if I select the C# formatting, so I don't get why Visual Studio doesn't.  Does anyone know of a VS2008 add-in that will enable this behavior?  Or some sort of hidden setting for it?  Edited to add: inserting regions is of course an option and it did already occur to me, but quite frankly, I shouldn't have to wrap things in a region that are already wrapped in braces... if I was going to edit the existing code, I would just refactor it to have better separation of concern anyway. ("wrapping" with new methods instead of regions ;) - but IMHO there has to be a simpler solution. If num-lock is on in the BIOS - why is it turned off during linux boot and/or KDE/Gnome/whatever startup? check the availability of the WCF Web Service I want to check the availability of the WCF web service i.c service is up or down through  the C# code. How to achieve that? Is there a way to include all the jar files within a directory in the classpath?  I'm trying java -classpath lib/*.jar:. my.package.Program and it is not able to find class files that are certainly in those jars.  Do I need to add each jar file to the classpath separately? One of the first things ever taught in a differential calculus class:   The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$.   This leads to a rather neat (and convenient?) chain of derivatives:   sin(x) cos(x) -sin(x) -cos(x) sin(x) ...   An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$  is at a maximum, $\sin x$ is zero and moving upwards.  But these "matching points" only work for multiples of $\pi/4$.  Let us move back towards the original definition(s) of sine and cosine:  At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle.  To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis.  The definition of $\cos x$ was then made the same way, but with adj/hyp and the X-coordinate, as we all know.  Is there anything about this basic definition that allows someone to look at these definitions, alone, and think, "Hey, the derivative of the sine function with respect to angle is the cosine function!"  That is, from the unit circle definition alone.  Or, even more amazingly, the right triangle definition alone.  Ignoring graphical analysis of their plot.  In essence, I am asking, essentially, "Intuitively why is the derivative of the sine the cosine?" Converting to upper and lower case in Java I've got a list of Python objects that I'd like to sort by an attribute of the objects themselves.  The list looks like:  &gt;&gt;&gt; ut [&lt;Tag: 128&gt;, &lt;Tag: 2008&gt;, &lt;Tag: &lt;&gt;, &lt;Tag: actionscript&gt;, &lt;Tag: addresses&gt;,  &lt;Tag: aes&gt;, &lt;Tag: ajax&gt; ...]   Each object has a count:  &gt;&gt;&gt; ut[1].count 1L   I need to sort the list by number of counts descending.  I've seen several methods for this, but I'm looking for best practice in Python. Wrong numeration in toc longer then one page How can I see the history of stories I've read on Apple News? Sometimes I'll be interrupted in the middle of reading a story on Apple News and later when I open the app, it didn't save the story I was previously reading. I want to be able to resume reading the article, how can I find it again? Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$. Prove the following: If $a \mid bc$, then $a  \mid \gcd(a, b)c$.   I tried to set $\gcd(a, b)$ to $b$ and used the fundamental theorem of arithmetic to prove that it is divisible by $a$, but I can't prove that $a \mid bc$, if and only if $a\mid b$ and $a\mid c$. Please help. Thanks. Guests see the Protected notice , somewhat displaced:    Logged in, all is fine:    It actually belongs there, at the bottom, but then it has some design glitches, or it needs to be placed at the top.  Verified on Firefox and Chrome stable. I know you can export the asset as a PSD and open it in Photoshop. Is there a more seamless way?  Update:  I'm an idiot.  Someone else created the logo, and the shadow was it's own layer on multiply and not a normal drop shadow effect. Is this correct: "Our listeners are what make X"?