How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas?  Note: I found that if it converges then it converges to ln(2)^2 but that doesn't help since I want to prove the it converges at first place.
How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas?
I have realized that to pronounce the plural form of words ending in -th, we have to drop the letter "h". e.g. Months is pronounced /mʌnts/. But I have never seen this point mentioned in any context. I just want to make sure that this rule is correct and can be always applied. Why do we use the object instead of the subject pronoun in constructions like "stupid me"? I'm trying to find out how come we say lucky me and stupid us rather than lucky I and stupid we. My understanding is that this is not a recent invention, but a relic from the distant past where it was a dative construction. However, I don't see the reason for using any case other than the nominative in this construction to begin with.        He went surfing. → Lucky him went surfing.   I got pulled over. → Stupid me got pulled over.   They bought a house. → Thoughtless them bought a house.      This is rather intriguing. After thinking about it for a while, I couldn't come up with any other language, Germanic or otherwise, where simply introducing an adjective would dictate a change of grammatical case. This appears to be an English peculiarity. How far can it be traced back? Can anyone comment on its background? Helllo!  For a project that I am doing, I need to render the my model to a sprite sheet. This is all fine, except that the model has a cloak, which uses a cloth animation. I need to smoothly transition from the end of the cloth simulation to the start, so it doesn't just jerk back, but I can't find any way of doing this.  Thanks in advance. connectedness of uniform topology Trouble with batch conversion of .png to .pdf using convert convert /home/bill/TempScan/*.png  myfile.pdf   gives error message:  convert-im6.q16: not authorized `myfile.pdf' @ error/constitute.c/WriteImage/1037.   Any help would be appreciated! Cannot determine size of graphic I'm trying to include graphics in my Latex-file, which I compiled with latex+dvipdf on OS X. Latex however returns this error: "Cannot determine size of graphic"  My graphic is exported from PowerPoint, so I have tried both .pdf and .png. I get the same errors for both.  Sample code that doesn't work:  \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure}   If I have to manually set some measurements of the image, please tell me how I can find out those measurements. I need the image to take up about 80% of text width and be centered. Thanks. What affects field upgrades in BF4? How to insert page number in Beamer navigation symbols? Is it necessary that an International cricket ground should has the boundary line in circle shape?Is it allowed by ICC to make it a square or ellipse or any other? What is the minimum and maximum area allowed for an international cricket ground? What is the maximum number of international cricket grounds that a country allowed to have? "404. That’s an error." when viewing from Google cache My site is getting 404. That’s an error. message when I click on cached from Google search results.  This is the error message from Google:  That’s an error.   The requested URL /search?q=cache:8kxLAgFrhZQJ:www.mysite.com.....&amp;cd=8&amp;hl=en&amp;ct=clnk&amp;gl=uk was not found on this server. That’s all we know.  What I did before I was getting this error is that about 10 days ago, I moved my hosting from Ixwebhosting to Gearhost (cloud based). I also added 301 redirect from non www to www.  I have no idea why I am getting this error. Please advise what might be the reason if there is a solution for it.  All of my pages are indexed by Google though. There's an election going on. What's happening and how does it work? Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places? Assessing approximate distribution of data based on a histogram Suppose I want to see whether my data is exponential based on a histogram (i.e. skewed to the right).   Depending on how I group or bin the data, I can get wildly different histograms.   One set of histograms will make is seem that the data is exponential. Another set will make it seem that data are not exponential. How do I make determining distributions from histograms well defined? Does the series $\sum\limits_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$ converge? Does the following series converge? $$\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$$   As $$\frac{1}{n\sqrt[n]{n}}=\frac{1}{n^{1+\frac{1}{n}}},$$ I was thinking that you may consider this as a p-series with $p&gt;1$. But I'm not sure if this is correct, as with p-series, p is a fixed number, right ? On the other hand, $1+\frac{1}{n}&gt;1$ for all $n$. Any hints ? Usage of "second/third/fourth ... last" In German there is a pattern for counting items from the end of a list. The last item is "das letzte", the one before is "das vorletzte", the one before that is "das vorvorletzte" and for each other item there's just another "vor" added.  While searching for an English translation for the German word "vorvorvorletzte" I came across   and  but none of them showed any usage pattern. Furthermore I didn't find any indication of which variant ("to", "from", no preposition) is used most nor to what area of the world those translations apply.  So I am still wondering: When there are items 1, 2, 3, 4 and 5, how can/will item 2 be called?   third from last? fourth from last? anteantepenultimate? fourth-last? fourth to last? the last but three item? the last but four item?   Is this a question of dialect/local use? I clicked on  to a comment in my email and it prompted a list of apps to open link. All browsers (Chrome, Firefox and etc) are in the list, but not SE app.  Most apps that can open site specific links get prompted for that app to open  the link, but not Stack Exchange links for some reason.    App Version: 1.0.95 Device Manufacturer: Asus Device Model: ASUS_Z010D   OS Version: 6.0.1 (WW_Phone-13.8.26.76-20161230) I want to know which sentence is correct. If i say     Guess, what it is?   or     Guess, what is it?   which one is correct? Example of set which contains itself I am trying to understand   How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself. I am considering trying to setup a  system. It seems like this would be pretty straight forward if I had two graphics cards or if I didn't need hardware based OpenGL acceleration. Is it possible to setup a multi seat system with only a single dual-headed graphics card? I have a character playing a multiclass Barbarian/Monk. Does the resistance from rage combine with Deflect Missiles to negate the damage and make the reaction attack?