Proving that Cauchy Summation converges
Proving that $\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$ converges
How many passes do you need to wipe/shred your files to make them not undeletable? Blank screen after installing nvidia restricted driver To consume REST services there is a lot of boilerplate code needed:   Build Apex Objects to model the responses Code to deserialize REST responses to those classes Handling of HTTP requests, responses and errors ...   Did anyone come up with patterns or best practices codified into an open source library or framework that one could just reuse in a similar project without reinventing the callout-wheel over an over again?  Like the zillions of trigger-frameworks that exist as repos on Github... I was playing around with some integrals and noticed that some integrals of the form:  $$I(a,b,c)=\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$$ Does have a closed form. I am trying to find for what constant $c$ will this work. In case you wonder why only $c$ is problematic, I will try to show by an example.  $$I=I(1,3,16)=\int_1^\infty \frac{\arctan(x+3)}{x^2+16}dx$$ Let's start by letting $x-1=t\,$ thus: $$I=\int_0^\infty \frac{{\arctan(\color{blue}{t+4})}}{t^2+2t+17}dt$$ With $\displaystyle{t=\frac{17}{y}\rightarrow dt=-\frac{17}{y^2}dy}$ $$I=\int_0^\infty \frac{{\arctan\left(\color{red}{\frac{17}{y}+4}\right)}}{\left(\frac{17}{y}\right)^2 +\frac{34}{y}+17 }\frac{17}{y^2}dy\overset{y=t}=\int_0^\infty \frac{{\arctan\left(\color{red}{\frac{17}{t}+4}\right)}}{t^2+2t+17}dt$$ $$2I=\int_0^\infty \frac{{\arctan(\color{blue}{t+4})+{\arctan\left(\color{red}{\frac{17}{t}+4}\right)}}}{t^2+2t+17}dt$$ $${\arctan(\color{blue}{t+4})+{\arctan\left(\color{red}{\frac{17}{t}+4}\right)}}=\arctan\left(\frac{\color{blue}{t+4}+\color{red}{\frac{17}{t}+4}}{1-(\color{blue}{t+4})\left(\color{red}{\frac{17}{t}+4}\right)}\right)$$ $$=\arctan\left(\frac{x^2+8x+17}{x}\frac{x}{-4(x^2+8x+17}\right)=\pi-\arctan\left(\frac14\right)$$ Above follows since the original integral is positive so we take $\arctan(-x)$ as $\pi-\arctan x $ and therefore getting a negative answer will not be an issue. $$I=\frac12 \left(\pi -\arctan\left(\frac14\right)\right)\int_0^\infty \frac{1}{t^2+2t+17}dt$$ Well, now the inner integral is not hard to compute and the final answer happens to be: $$I=\frac12 \left(\pi -\arctan\left(\frac14\right)\right)\frac14\arctan\left(\frac{t+1}{4}\right)\bigg|_0^\infty =\frac{\pi^2}{16}-\frac{3\pi}{16}\arctan\left(\frac14\right)+\frac18\arctan^2\left(\frac14\right) $$ There are more examples that I found by checking and try such as:  $$I(1,2,9)=\int_1^\infty \frac{\arctan(x+2)}{x^2+9}dx$$ $$I(2,1,6)=\int_2^\infty \frac{\arctan(x+1)}{x^2+6}dx$$ $$I(2,2,13)=\int_2^\infty \frac{\arctan(x+2)}{x^2+13}dx$$ And so on... All those can be solved by the same method: First substitute $x-a=t$, then let $t=\frac{\alpha}{y}$, where $\alpha$ is the "free of x" coefficient from the denominator.   The problem is that I tried more than $100$ combinations to get those integrals which is not that nice. How can we "smartly" find $c$ so that $I(a,b,c)$ is evaluable by symmetry? Or put in other word what should be $c$ if one wants to compute by symmetry $I(7,13,c)$? Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.    The problem can be solved by putting every element from $\mathbb{Z}_{11}$ and checking if there is any root.  $0^2 + 0 + 4 = 4 \\ 1^2 + 1 + 4 = 6 \\ 2^2 + 2 + 4 = 10 \\ 3^2 + 3 + 4 = 5 \\ 4^2 + 4 + 4 = 2 \\ 5^2 + 5 + 4 = 1 \\ 6^2 + 6 + 4 = 2 \\ 7^2 + 7 + 4 = 5 \\ 8^2 + 8 + 4 = 10 \\ 9^2 + 9 + 4 = 6 \\ 10^2 + 10 + 4 = 4 $   But this process is very long one. I am seeking a short method by which I can solve such problem.   Now suppose that it is reducible, then the factors should be linear. So how can we solve the problem based on this kind of observation? Any help will be great. Thanks. What are good books/other readings for elementary set theory? I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to your mind that would be useful for an undergrad math student who hasn't taken a set theory course yet?   Thanks a lot for your suggestions! What is the difference between lands and mana? How to ensure OpenVPN connection uses specific DNS? I'm using OpenVPN through Tunnelblick on MacOS X Lion.  I need to set specific DNS (with local IP, which works only when VPN is up) for the duration of this VPN session only.  I do not have access to the OpenVPN server configuration. Only client config. Also, DNS from the server doesn't work.  So it works like this: I connect to VPN, go the Network preferences and manually set DNS. After VPN is disconnected, I switch back to default. It works, but it needs to be automatic.  After some exploration I found that OpenVPN up- and down- scripts might help me with that. Unfortunately, I haven't found any specific documentation about how exactly it can be done.  How it can or can't be done? Any advice would be appreciated! Add ability to flag answers as duplicates Uninstalling Ubuntu from dual boot / changing bootloader Do I have to care about bad boxes? When I compile my PDF from LaTeX source, do I have to care about all the messages (currently 28) about 'bad boxes'?  The PDF seems fine to me. 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. An interesting exercise about converging positive series, involving $\sum_{n\geq 1}a_n^{\frac{n-1}{n}}$ When is it OK to shrink a Database? I know shrink is the devil: It reverses page order and is responsible for skin cancer, data fragmentation, and global warming.  The list goes on... That being said, say I have a 100 GB database and I delete 50 GB of data -- not on one table, but a general pruning of old data on a database wide level, covering 90% of the tables -- does this constitute an appropriate use case for shrinking the database?  If not, what are the appropriate steps to take to clean house after removing such a high percentage of data from a database?  I can think of two: Rebuild Indexes and Update Stats. What else? So I understand that this has happened in the past: , but all of the &quot;duplicate&quot; posts are marked as  so here is a new bug report: On the , I am listed twice:  While that's fun and all, it isn't exactly right. Does the Force wane as you age? @Valorum said     ...not to mention the sheer strain that comes from continually using the Force at an advanced age.   To my feeble mind, this doesn't make sense. Approaching 900 years old, Yoda is still as strong as any whippersnapper in terms of Force power and usage. Despite his old age, Obi Wan shouldn't have any problems with using his Force powers.  This got me to thinking, does the Force begin to leave you as you age? Is it a flow of Force power that goes down based on (I hate to say this) a decrease in midichlorian count as you approach your expiration date? "Coming soon" pages & SEO I am setting up my personal website, and I have designed a simple "Coming Soon / Under Construction" page which basically has nothing but a small notice for the users.   What kind of SEO can I make on this page such that it won't harm the full website's ranking later on? How do I suppress the "New release '12.10' available" message? Homework - set theory infinite union A question from my homework I'm having trouble understanding. We are given:  $A(1) = \{\varnothing\}$, $A(n+1) = A(n)\cup (A(n)\times A(n))$  $A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots$ to infinity  The questions are:  1) show that $A\times A \subseteq A$  2) Is $A \times A = A$?  Thank you for your help.  I've tried writing $A(2)$ but it gets really complicated and I'm having trouble understanding what the sets are. Let alone solve the question. How to apply a macro to each column of a table