I was playing around with the function $\dfrac{1}{1+x^2}$, and knowing that the integral over $(0,\infty)$ was $\dfrac{\pi}{2}$, I was hoping to see if there was some neat pattern to determining the value of the following generalized form: $$I=\int_0^\infty \frac{dx}{1+x^n}$$ for natural $n\ge2$. I ran some computations in Mathematica and got a consistent result of $\dfrac{\pi}{n}\csc\left(\dfrac{\pi}{n}\right)$, which also happens to be equivalent to $\dfrac{1}{n}\Gamma\left(\dfrac{1}{n}\right)\Gamma\left(1-\dfrac{1}{n}\right)=\mathrm{B}\left(\dfrac{1}{n},1-\dfrac{1}{n}\right)$.  Is there some neat trick I can use to get this result? I've considered integrating along a contour, but I'm somewhat rusty with complex methods. Also, the result was posted previously .  I tried something like this instead: $$\frac{1}{1+x^n}=\begin{cases}\displaystyle \sum_{k=0}^\infty(-1)^kx^{nk}&amp;\text{for }|x|&lt;1\\\\ \displaystyle\sum_{k=0}^\infty (-1)^k x^{-n(k+1)}&amp;\text{for }|x|&gt;1\end{cases}$$ then integrating along the respective intervals $(0,1)$ and $(1,\infty)$.  Here's my attempt: $$\begin{align*}I &amp;=\left\{\int_0^1+\int_1^\infty\right\}\frac{dx}{1+x^n}\\\\ &amp;=\sum_{k=0}^\infty \left\{\int_0^1 (-1)^kx^{nk}\,dx+\int_1^\infty (-1)^kx^{-n(k+1)}\,dx\right\}\\\\ &amp;=\sum_{k=0}^\infty \left\{(-1)^k\left[\frac{x^{nk+1}}{nk+1}\right]_{0}^{1}+(-1)^k\left[\frac{x^{-n(k+1)+1}}{-n(k+1)+1}\right]_1^\infty\right\}\\\\ &amp;=\sum_{k=0}^\infty \left\{\frac{(-1)^k}{nk+1}-\frac{(-1)^{k}}{1-n(k+1)}\right\}\\\\ &amp;=\left(1-\frac{1}{1-n}\right)+\left(-\frac{1}{n+1}+\frac{1}{1-2n}\right)\\ &amp;\quad\quad+\left(\frac{1}{2n+1}-\frac{1}{1-3n}\right)+\left(-\frac{1}{3n+1}+\frac{1}{1-4n}\right)+\cdots\\\\ &amp;=1-\frac{2}{1-n^2}+\frac{2}{1-4n^2}-\frac{2}{1-9n^2}+\cdots\\\\ &amp;=\sum_{k=0}^\infty \frac{(-2)^k}{1-(kn)^2}\end{align*}$$ However, this series diverges by the ratio test, so I'm not sure if I made a mistake in my calculation or used some flawed reasoning.
I am trying to evaluate the following:  $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a&gt;1$ and $a \in \mathbb{R}$  Any help will be much appreciated.
VS10 debugger error: Test host process exited unexpectedly Best Algebraic Geometry text book? (other than Hartshorne) How to integrate PHP forum system with new Django website I figured out how to use a normal image as the background while using an HDRI for environmental lighting. But I can't figure out how to use a different HDRI as the background.  Please note I already read this answer:  but this appears to be different for Eevee in 2.8.  Thanks! Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$ Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$.  Ok I'm finally understanding what each of these things mean.. but I'm running out of ideas on how to actually show it without using arbitrary vector space examples like Rn. Since sets and subspaces are kind of new its hard to figure out how to write correct proofs with them.   I can see how if $V = \mathbb{R}^3$ and $S = \{(1,1,1)\}$ and $W = \{(1,1,1), (1,1,2)\}$ then $\mathrm{span}(S) = R1(line), span(W) = R2(plane)$ and that R1 is a subspace of R2... im just having trouble showing this officially when all vector spaces and sets are completely in general....  I could get as far as writing out $S=(u_1,u_2,\dots,u_n)$, $u_i \in V$ and same with $W$, but I can't really figure out what else to do in this sort of proof. Bachelor thesis or Bachelor's thesis I recently bought the whole PC series of Saints Row and after I began the third, after I leveled up, I saw that most of the power upgrades were already done for me and many clothing items were already bought.  I know the game is quite easy  economically but I want to start a game in which I can actually level up, not sideways.  Is it possible to begin a game in which (preferably) everything is locked?  I probably have all the downloadables of the game. Is it possible to make my laptop's screen act as a second monitor for my desktop computer? In this movie there are scientists who (spoilers ahead)       ...are creating a simulation. One scientist starts seeing odd looking characters but no one else sees them. He asks around and more and more characters are missing or don't know about the characters in the story he knows about.  He finds out that he is part of a simulation. He finds out that the characters only he remembers have been deleted. He also finds out that there are characters from his own simulation that have broken free of his simulation and are in his world.    Hints: I remember this being an older film (70's?); it might have been foreign film possibly with subtitles (not Hollywood); it might have been part of the Criterion collection; I'm pretty sure I streamed it from Hulu. Sorry, I can't remember more details. Why do video cables (DVI, VGA) still screw in? Density of finite rank operator in compact operators on Hilbert spaces Is the sign of principal components meaningless? Does the sign of the vectors returned by PCA have any significance after I center my variables? Avoiding getting suspended for voting irregularities? Many times when I read threads, I keep on upvoting for the reason being that if someone has put an effort to write a big answer, s/he/it should get some rewards for it.  I recently saw someone getting suspended for voting irregularities.   So, I was wondering if I use the total amount of votes allowed per day, daily, will I be suspended? No more than 10,000 results are visible in search If you make a search for something that returns more than 10,000 items in result. For example, search for  return 161,338 results on MSE. But if you select a page where 10,000+ result should be shown:        In the edge case  shows zero results:        But this is actual for any page where per_page * page_number is more than 10,000.  Though this does not happen for Stack Overflow in English. It looks like all other sites are affected.  Related bug report on ruSO.meta: How do I get attention for old, unfixed bug reports and feature requests without official responses here on Meta? I posted a question here on Meta Stack Exchange, or on a per-site meta, regarding a bug I found in the system. I got no comments, or a few comments stating that this is a legitimate bug (and not by design). However, it's been quite a long time, and no official response was ever posted there.  Or, I had an idea for a new feature, but realized it's already requested, either by searching or posting a request myself only to have it closed as a duplicate of the existing one. It's been a while since the original request was filed, but no official response has been made.   I'm aware that staff members are , but either my post isn't about one of the indicated topic areas for which they're accepting escalation requests for that month, or I already flagged to request escalation but it was declined, so the instructions there don't apply for my post.  What can I do to draw attention to meta questions for the SE team, so they can give an official response, and possibly implement the bug fix or feature?  Note: This is different from other general questions which ask for how to get attention for unanswered questions, since the procedures on meta sites are different, and getting an official response isn't necessarily the same as getting a good answer. Lightning Web Components: navigate to the 'new' object page passing in specific record type Id? I am using LWCs to display lightning cards in a list.  The list shows information about Programs.  Each Program card has a button, and this button needs to take the user to the 'new object' screen of the Application object.  Here is my code:   applyToProgram(event) {     // The recordTypeId is coming from the .html     // I have verified that the value is correct.     var rtId = event.target.id;      // How do I use rtId below?     this[NavigationMixin.Navigate]({         type: 'standard__objectPage',         attributes: {             objectApiName: 'Application__c',             actionName: 'new',             nooverride: '1',             recordTypeId: rtId  // This does not work         }     }); }   This is not working.  The user is presented with the 'new object' screen for the object's default record type and NOT the one specified in rtId.  I tried putting 'nooverride' and 'recordTypeId' in the state:{} argument, and I also renamed 'recordTypeId' to just 'recordType'.  None of these worked.  Can anyone help me with the correct syntax?  Is this even possible in LWCs?  Thanks, Jeff How to merge columns in a table? I have this table, but wanted to format the first row. I like to match the two columns of first row, and last two columns of the same first row. Any suggestion on how to do that?  \begin{table}[!h]  \caption{Comparison of percentages.} \begin{tabular}{lclclclclc} \hline \hline  Mode &amp;  Var  &amp;  Cum\\ \hline {}       &amp; EF   &amp; CHF    &amp; EF2   &amp; CHF2\\ 1   &amp;  17.5 &amp; 19.1   &amp; 17.5  &amp; 19.1\\ 2   &amp;  11.8 &amp; 12.7   &amp; 29.3  &amp;  31.9\\ 3   &amp;  6.6  &amp;  5.6         &amp; 35.9    &amp;  37.4\\ \hline \end{tabular} \end{table} How can I use the MS JDBC driver with MS SQL Server 2008 Express? My configuration:     windows XP SP3 JDBC 2005 MS SQL Server 2008 Express, exposed via tcp/ip on port 1433 sqljdbc.jar in class path    I tried:  try {     Class.forName("com.microsoft.sqlserver.jdbc.SQLServerDriver").newInstance();     con = DriverManager.getConnection("jdbc:microsoft:sqlserver://localhost:1433/SQLEXPRESS2008;databaseName=Test;selectMethod=cursor", "sa", ""); } catch (Exception e) {     e.printStackTrace(); }   But it always throws an exception:  java.sql.SQLException: No suitable driver   I also tried the following urls:  localhost:1433/SQLEXPRESS2008  localhost/SQLEXPRESS2008  localhost   Same results.  Any help? Show that if $a$ is an odd integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$ Show that if $a$ is an odd en integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$.  I understand that since $a$ is not divisible by $2$ but $b$ is, the gcd of $a$ and $b$ also can't be divisible by $2$ but I'm getting stuck using this to help me. How to access namespace which is part of different project?