Evaluate the sum $1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}+\ldots$ Evaluate the sum $1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}+\ldots$  Been looking for a way to arrive at a solution. Would really appreciate some help here.
Evaluating the sum $\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$ I am tasked to evaluate the sum  $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$$ Using contour integration. This is what I've done so far.   Let $C_{N}$ be the square defined by the lines $x=\pm(N+\tfrac{1}{2})\pi$ and $y=\pm(N+\tfrac{1}{2})\pi$. Let $f=\tfrac{1}{z^{2}\sin(z)}$. I was able to prove that  $$\int_{C_{N}}\dfrac{1}{z^{2}\sin(z)}dz=2\pi i\left[\dfrac{1}{6}+2\sum_{n=1}^{N}\dfrac{(-1)^{n}}{\pi^{2}n^{2}} \right]. $$  By Cauchy's residue theorem. So now if I could prove that he integral converges to zero then I would be done. The problem is that I can't seem to get an upper bound on the integral. I've having trouble with the fact that $|f|$ is unbounded within $C_{N}$ so ML inequality doesn't help me. Any hints would be great. Thanks.
Here's a screen capture from a site where I can view /site-analytics:    The orange "all posts" curve should be strictly larger than the blue "questions" curve, but they are mistakenly displaying the same data. Next badge select dialog broken The "Select your next badge" dialog box is not displaying correctly. It looks like this:    Freehand circles, as requested:    It seems to be an issue with box-sizing, because when it's content-box, it looks fine. I think the width and height rules for .popup-badges .all-badge-progress .badge-progress should be increased a little, for example:  .popup-badges .all-badge-progress .badge-progress {   width: 214px;   height: 96px; }   Tested on the latest Chrome and Firefox on Linux Mint. Is it possible to install an old version of Mac OS on a new Mac? Raised index of partial derivative I am having a really hard time wrapping my head around component notation for tensor fields. For example, I do not know exactly what the following expression means $$\partial_\mu\partial^\nu \phi, \tag{$\#$}$$ where $\phi$ is a scalar field. On the one hand $\partial^\nu=g^{\lambda\nu}\partial_\lambda$ where $g_{\mu\nu}$ is the Minkowski metric, and hence we could write explicitly $$\partial_\mu\partial^\nu \phi=\sum_{\mu,\nu,\lambda}g^{\lambda\nu}\partial_\mu \partial_{\lambda}\phi=\sum_{\mu,\nu}\partial_{\mu}\partial_{\nu}\phi=\partial_\mu\partial_\nu\phi. \tag{$*$} $$ On the other hand, we may think of $\partial_\mu\partial^\nu=g(\partial_\mu,\partial^\nu)=\delta_\mu^\nu;$ so that $\partial_\mu\partial^\nu\phi=\phi?$ Maybe? I am actually not sure of what this would mean. I am really confused. Any help is appreciated. Edit: To give context of where this expression comes from: I was computing the Lagrangian $$\mathcal L=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu \phi) $$ considering an infinitesimal spacetime translation $x^ \mu\to x^\mu-\alpha a^\mu$. The scalar field thus transforms like $\phi(x)\to \phi(x)+\alpha(\partial_\mu\phi(x))a^\mu.$ Plugging thins into the Lagrangian yields the term I am referring to. Edit 2: The change in placement of indices are actually my doubts. I try to elaborate. I do not have any background in using indices to talk about tensors. I am used to interpret the expressions $\partial_\mu$ as the local vector field defined in some chart (local coordinates). I think about vector fields $X$ as abstract section of the Tangent bundle, which restricted to local coordinates can be expressed as $X=X^\mu\partial_\mu$. In the context of QFT, as far as I understand, the symbol $\partial_\mu$ denotes  $(\partial_t,\nabla)$ in the local coordinates $(t,x,y,z)$. So that $\partial_\mu\phi=(\partial_t \phi,\partial_x \phi,\partial_y\phi,\partial_z\phi)$. This was supposed to be my justification on why I wrote the summation on $\mu$ and $\nu$ in $(*)$, but now I note that this only applies when $\mu$ or $\nu$ appear twice, indicating the scalar product; which leads me to the last remark. I think of $g_{\mu \nu}$ as the component of the matrix $$g=\begin{pmatrix} 1&amp;0&amp;0&amp;0\\ 0&amp;-1&amp;0&amp;0\\ 0&amp;0&amp;-1&amp;0\\ 0&amp;0&amp;0&amp;-1\\ \end{pmatrix}$$ which represents the pseudo-Riemmanian metric, which by definition acts on tangent vectors, i.e. linear combinations of the $\partial_\mu$ applied to a point. This is where my doubt comes, in which was the right way to interpret the notation; in particular what is the expresion $(\#)$ in explicit coordinates? Are there any simple rules for choosing the definite vs. indefinite (vs. none) article? I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know. Excel: I need a formula that will count months between two dates. If it's 1-15 it should be a half month. Any ideas? So I am trying to find a formula for excel. Bascically I have two dates and I need to know the number of months between them. But to add complexity, I need a rule in there that if the earlier date is the 1st to 15th it will be a whole month but if it is 16th to end of the month it will be a half month.  For example: Date 1: April 4, 2012 Date 2: December 31, 2012 Answer: Number of months would be 9 months  If Date 1 was April 16, 2012, with the same Date 2 of December 31, 2012, then the number of months would equal 8.5.  Can anyone help me with this one? You help would be greatly appreciated. many thanks Show the series $a_n/(1+a_n)$ converges absolutely I apologize for the subjective "best" keyword.  My friend and I have started creation of a 2D adventure game. It will be top-down in the style of pokemon or zelda (just the perspective). We have been discussing methods of creating a large world-map that the player can traverse without straining the memory capabilities of our machine.  Our first impulse was to create a large map and a circle around the player in which content will be loaded. We figured this won't hold for long and decided to partition the map into sections. First we had four large sections, but realized we could simply break it down into many tiny sections.  I played some Zelda from the SNES and saw that, during the shift of a map, the content could be loaded just then. What I mean is, instead of just checking a rectangular area for data to load, we simply section the map into many tiny chunks that load and de-load data as we move from map-portion to map-portion.  Today, he told me he wanted to create a simply 2D array map[WIDTH][HEIGHT] that contains data about every grid in the game and is a constant save-to-disk operation for data that we don't need.  I'm not sure about these ideas and thought that I might as it here. Any links, resources or tutorials on the subject would be greatly appreciated as well as direct answers to our question on how to do it efficiently. Where are the Unity desktop wallpapers located? number of prime ideals in algebraic integers I have a hard time formulating proofs.  For this problem, I  can see that if $n$ is equal to $8,$ this statement is true.  $(\mathbb{Z}/8\mathbb{Z})^{\times}$ includes elements:  $1,3,5,7$, and all of these are roots of $1-x^2 \pmod 8.$  And obviously $8$ divides $24.$  But how do I prove this without depending on number calculations and only using theorems?  Help Please?  I need a step by step walk through of how to do this proof and what theorems would be appropriate to use. Functional equations leading to sine and cosine Why do I sometimes see a downloading 'pie' icon when viewing photostream photos? it normally shows in the bottom corner as a sort of 'pie' that fills up as the picture is downloaded, similar to the 'app downloading' pie when viewing a downloading app on the home page/s  prior to this the picture resolution is terrible How to align checkboxes and their labels consistently cross-browsers URLs are linked despite being inside a code span Markdown source:  `http://google.com`  - `http://google.com`     - `http://google.com`   Rendered result:  http://google.com   http://google.com   http://google.com    Note that the third code span is a link. Adding a newline before the list item makes the link disappear. Another simple series convergence question: $\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the convergence of $\displaystyle\sum\limits_{n=3}^\infty\frac 1{n\ln (n\ln 2)}$. Am I on the right path, and how do I proceed from here? Asymptotic of a sum involving binomial coefficients Deriving the conditional distributions of a multivariate normal distribution How did the letter Z come to be associated with sleeping/snoring? How to get Url Hash (#) from server side