Find the sum of the series $ \ \sum_{n=1}^{\infty} \frac{1}{n^2+l^2} \ $ , where $ \ l=constant \ $   Answer  The given series is convergent clearly .   $ \ \sum_{n=1}^{\infty} \frac{1}{n^2+l^2} \\ = \frac{1}{1^2+l^2}+\frac{1}{2^2+l^2}+......+\frac{1}{n^2+l^2}+........ \\ = \frac{1}{l}  [ d \tan^{-1}(\frac{1}{l})+ d \tan^{-1}(\frac{2}{l})+........+ d \tan^{-1}(\frac{n}{l})+.....] \\ = \frac{1}{l} \lim_{n \to \infty}  \sum_{k=1}^{n}[d \tan^{-1}(\frac{1}{l})+ d \tan^{-1}(\frac{2}{l})+........+ d \tan^{-1}(\frac{k}{l})]    \\ = \frac{1}{l} \ d (\lim_{n \to \infty}  \sum_{k=1}^{n}[ \tan^{-1}(\frac{1}{l})+  \tan^{-1}(\frac{2}{l})+........+  \tan^{-1}(\frac{k}{l})]) $  but now i  can't proceed to find the sum of the series.   Help me out.
Does anyone know the general strategy for summing a series of the form:  $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$  where $a$ is a positive integer?  Any hints or ideas would be great!
How do I write a URL in Markdown on Stack Exchange? How do I remove the Me-User-Indicator from the Panel? How do I remove the Me-User-Indicator from the Panel?  The indicator contains two items: the written user name and the icon of a human. Asymptotes in a plot I'm using TikZ to plotting some functions for a document. For example, I want to plot 1/(x+1). So I did something like this:  \begin{tikzpicture}[&gt;=stealth,x=3mm,y=3mm] \draw (0,5) node[above]{\fimg $y=f(x+1)=\dfrac{1}{x+1}$}; \clip (-5,-5) rectangle (5,5); \ejes[1]{-4}{4}{-4}{4} \draw[samples=100] plot(\x,{1/(\x+1)}); \end{tikzpicture}   where \ejes[style]{xmin}{xmax}{ymin}{ymax} is command, that I have defined before to draw the axes (join with \fimg). My expected output for the code below was just the function, but I get this:    So, TikZ automagically put the vertical asymptote at x=-1. That isn't a bad thing, but now I want to have some control over it. For example I want dotted lines for vertical asymptotes. How can I do that? Why do non-constant periodic functions have no limit at infinity? Algorithm to match one input file with given numbers of file There are 11 letters in the word.   M - 1 I - 4 S - 4 P - 2  so the number of different permutations is $\dfrac{11!}{1! 4 !4!2!}$  Is this correct solution? 'as compared to' vs 'as compare to' Resizing an iframe based on content I am working on an iGoogle-like application. Content from other applications (on other domains) is shown using iframes.   How do I resize the iframes to fit the height of the iframes' content?  I've tried to decipher the javascript Google uses but it's obfuscated, and searching the web has been fruitless so far.  Update: Please note that content is loaded from other domains, so the  applies. Gradient borders Is there a word to describe words that can be spelled different ways? Is there a word in English to describe words that can be spelled more than one way?  For example:   colour and color favourite and favorite   However, this shouldn't be limited to words spelled differently in different dialects, though I'm not sure what other case there would be. Check if one list contains element from the other Why does Spring MVC respond with a 404 and report "No mapping found for HTTP request with URI [...] in DispatcherServlet"? Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps.... What are suitable TV sizes and viewing distances for couch surfing on a 1080p LCD HDTV? Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$ How do I change my timezone to UTC/GMT? Currently when I execute the date command from bash I get the following displayed  Thu May 17 12:21:00 IST 2012  Below is my timezone config based on executing dpkg-reconfigure tzdata     Current default time zone: 'Europe/Dublin'   Local time is now: Thu May 17 12:32:52 IST 2012.   Universal Time is now: Thu May 17 11:32:52 UTC 2012.     Any idea how I change from IST to UTC/GMT? A is similar to $A^k$, then each eigenvalue of $A$ is a root of unity Let $A \in \mathbb{C}(n,n)$ and $k \geq 2$ be an integer such that $$A \sim A^k$$. Show that if $A$ is non-singular then each eigenvalue of $A$ is a root of unity.   Attempt: Since $A \sim A^k$, $$PA = A^kP$$ where $P$ is an invertible matrix. Since $A$ is invertible, $0$ cannot be an eigenvalue of $A$. Suppose $$Av = \lambda v \quad v \neq 0$$ then $$PAv = \lambda Pv$$ $$\therefore A^k(Pv) = \lambda (Pv) $$ but this implies that $Pv$ is an eigenvector of $A$. But the eigenvalues of $A^k$ are $\lambda^k$ $$\therefore \lambda^k=\lambda$$ which gives the conclusion required.   My questions is: Is the logic correct? If so, Am I missing any details? If not, then how could I approach this?   Thanks! Explanation behind Second Derivative of a Parametric Equation Formula I am looking to find out how the second derivative formula works.  I can blindly apply it but I don't have a grasp of what is going on or why. The first derivitive has the formula $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. I told this was because if you divide them you cancel out the dts and get dy/dx. The second derivative has the formula $\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) = \frac{ \frac{d}{dt}\left(\frac{dy}{dx}\right) }{\frac{dx}{dt}}$. Is this something that can be explained with only a knowledge of calculus or is this something I would have to take on faith until I learn more. Any help would be greatly appreciated. Where does the phrase "on the lam" come from? According to the Online Etymological Dictionary, lam means:     "flight," as in on the lam, 1897, from a U.S. slang verb meaning "to run off" (1886), of uncertain origin, perhaps somehow from the first element of lambaste, which was used in British student slang for "beat" since 1590s.    Does anyone know of any other explanations? Pluralizing abbreviations where the noun is not the last word