How to find $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ using complex analysis? Consider the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$. How to find its value using complex analysis?  I am considering $f(z)=\sum_{n=1}^{\infty} \frac{z^n}{n^2}$. It is absolutely convergent in $|z|&lt;1$. So it is holomorphic and can be differentiated termwisely, namely $f'(z) = \sum_{n=1}^{\infty} \frac{z^{n-1}}{n}$, thus $zf'(z)=\sum_{n=1}^{\infty} \frac{z^n}{n} = \log (1-z)$ where $\log$ takes principal branch such that $\log1=0$. Thus we have $$ f'(z) = \frac{\log (1-z)}{z} $$ I cannot proceed now. Is it possible to use complex analysis theories to tackle this?  Thank you for any help!
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.
I'd like to get a notification when a question that I've voted to close (but isn't closed yet) is edited Hello I am Indian Citizen currently studying in Australia and I am on a student visa. In the month of November 2019 I am travelling to India for vacations and I booked a ticket from Melbourne to Bangkok and than form Bangkok to India with a layover time of 7 hours in Bangkok and I am not going to leave the airport but my flights are different, so do I need to get and Airport Transit Visa in Bangkok. MySQL Error 1093 - Can't specify target table for update in FROM clause This morning, I had eggs for breakfast, and I was looking at the pieces of broken shells and thought "What is the surface area of this egg?" The problem is that I have no real idea about how to find the surface area.    I have learned formulas for circles, and I know the equation for an ellipse; however, I don't know how to apply that.   The only idea I can think of is to put an egg on a sheet of paper and trace it, and then measure the outline drawn, and then try to find an equation for that ellipse and rotate that about the $x$-axis.  Now, my problem is how I can find the equation of the ellipse from the graph, and will my tracing method really be the edge of the egg?  Also, can I use the ? Will I have to use some techniques to solve the integral that are not covered in the ?  There has to be a better method for finding the surface area. Please, help me understand how to find the surface area of an egg; i.e., how to use my mathematical knowledge for something other than passing exams. Do descriptive statistics have p-values? I want to find the interval of convergence of $$\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(2x)^n$$ By the root test, $\sqrt[n]{|(\frac{n}{n+1})^{n^2}(2x)^n|}=|(\frac{n}{n+1})^{n}(2x)|$  And $(\frac{n}{n+1})^n=(1-\frac{1}{n+1})^n\to\frac1e$ as $n\to\infty$  So I've got an open interval(of convergence) s.t. $|\frac1e 2x|&lt;1\implies |x|&lt;\frac e2$.  Now I need to figure out whether the series is convergent or not at the end points, $x=\frac e2, -\frac e2$.  So there are two series I need to deal with, namely, $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}e^n$ and $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(-1)^ne^n$.  Suppose $(\frac{n}{n+1})^{n^2}$ is increasing. Then $$(\frac{n}{n+1})^{n^2}&lt;(\frac{n+1}{n+2})^{(n+1)^2}\iff 1&lt;(\frac{n+1}{n+2})^{(n+1)^2}(\frac{n+1}{n})^{n^2}$$ $$\iff 1&lt;(1-\frac{1}{n+2})^{2n+1}(1+\frac{1}{n^2+2n})^{n^2}$$  But $(1-\frac{1}{n+2})^{2n+1}\to\frac{1}{e^2}$ and $(1+\frac{1}{n^2+2n})^{n^2}&lt;(1+\frac{1}{n^2})^{n^2}\to e$  So I have $1&lt;\frac1e&lt;1$, which is a contradiction. Hence $(\frac{n}{n+1})^{n^2}$ is not increasing. But I don't know if it's decreasing. (is it true that $(1+\frac{1}{n^2+2n})^{n^2}\to e$?)  If $(1+\frac{1}{n^2+2n})^{n^2}\to e$ is true, then $(\frac{n}{n+1})^{n^2}e^n$ converges to some positive number, so $\sum(\frac{n}{n+1})^{n^2}e^n$ diverges. But I don't know if it's decreasing: can't apply the alternating series convergence test(Leibniz's criterion).  So, can you help me with the two end points? What is the difference between a predicate and function? I need to to understand the difference between predicates and functions in the context of Clasual Form Logic in order to define the Herbrand universe.  If I have p(x) :- q(f(x)) would I be right in saying that p and q are predicates while f is a function because it is "nested"?  By this thinking then if I have p(x) :- q(x) both p and q are predicates and I have no functions?  If this is incorrect then how can I tell the difference between a predicate and function? Difficulty in finding a counterexample Why does a helium filled balloon move forward in a car when the car is accelerating? This poor creature is taken aboard the spaceship and the humans do not know that it is intelligent.  The alien can repeat what it hears, e,g, "pass the sugar".  It is able to escape at night and it communicates messages to its fellow aliens.  It directs that they turn up the heat of the atmosphere (somehow they can do this) so as to force the humans to leave.  The alien protagonist states that it will die w/ the humans or leave with them. LM317 Heat sink, how big? How to use Whatsapp mobile application on Ubuntu Touch? In my Android I have "Whatsapp" application running. But mobile freak I am not, and specially the mobile keyboards.  So how can I run my same application, same account from my Ubuntu desktop PC?   Follow up:  Just to do keyboard routing, i have used . Which allows me to takeover the Android input fields access via web link to type in my PC keyboard, and then it re-route the keys to Mobile.  Features:   Allows me to copy and paste large texts to Android input fields without using the keyboard of Android but simply using my Desktop PC, via web link http://android.ip.address.goes.here:port/ Allows me to have key submit which is similar to key press of mobile set itself   At-least works without having mobile in rooted. Hope it helps you!! Zorn's lemma and maximal ideals Let's consider two statements: Zorn's lemma and theorem about existence of maximal ideals in commutative ring with $1$.  It's easy to prove that Zorn's lemma implies existence of maximal ideals.  I wonder if the converse proposition is true, i. e., does existence of maximal ideals implies Zorn's lemma? My approach was to construct a ring structure on an ordered set which satisfies Zorn's lemma condition such that existence of maximal ideal implies existence of maximal element in the set, but I didn't succeed. Irrationals: A Group? I understand that the set of irrational numbers with multiplication does not form a group (clearly, $\sqrt{2}\sqrt{2}=2$, so the set is not closed). But is there a proof or a counter-example that the irrationals with addition form (or do not form) a group?  Thank you!  Edit: In particular, I am wondering if the set is closed with respect to addition. How can I know if a branch has been already merged into master? I have a git repository with multiple branches.   How can I know which branches are already merged into the master branch? Orthogonal Projection onto the $ {L}_{\infty} $ Unit Ball What is the Orthogonal Projection onto the $ {\ell}_{\infty} $ Unit Ball?  Namely, given $ x \in {\mathbb{R}}^{n} $ what would be:  $$ {\mathcal{P}}_{ { \left\| \cdot \right\| }_{\infty} \leq 1 } \left( x \right) = \arg \min_{{ \left\| y \right\| }_{\infty} \leq 1} \left\{ {\left\| y - x \right\|}_{2}^{2} \right\} $$  I managed to get an answer using the . Yet I would be happy to see if someone can derive the answer directly.  Thank You. How did "sinister", the Latin word for "left-handed", get its current meaning? Sinister is the Latin word for left-handed.  What evolution of meaning turned left-handed into evil and threatening? Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$? I looked at the graph of $f(x,y)=(xy)^{2/3}$ at the origin, and no matter how far I zoom in it has four corners along $x=\pm y$, so it doesn't look like it is differentiable as $z=0$ doesn't look like a tangent plane.  I also found that its partial derivatives are not continuous at $(0,0)$, e.g.  $$ \lim_{(ky^2,y) \to (0,0)} \frac{\partial f}{\partial x} = \lim_{(ky^2,y) \to (0,0)} \frac{2}{3}x^{-\frac{1}{3}}y^{\frac{2}{3}} = \frac{2}{3}k^{-\frac{1}{3}} $$ (although $\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0$)  I also tried to prove/disprove its differentiability by definition, and for it to be differentiable I need $$ \lim_{(x,y) \to (0,0)} \frac{f(x,y)-0}{\sqrt{x^2+y^2}} =0 $$ but then I got stuck. Am I even on the right track?  [UPDATE] I need to do the same thing for $f(x,y)=(xy)^{\frac{1}{3}}$... I think the answer is that $(xy)^{2/3}$ is differentiable but $(xy)^{1/3}$ is not, according to my professor but I'm not sure What's the best way to call a generic method when the type parameter isn't known at compile time, but instead is obtained dynamically at runtime?  Consider the following sample code - inside the Example() method, what's the most concise way to invoke GenericMethod&lt;T&gt;() using the Type stored in the myType variable?  public class Sample {     public void Example(string typeName)     {         Type myType = FindType(typeName);          // What goes here to call GenericMethod&lt;T&gt;()?         GenericMethod&lt;myType&gt;(); // This doesn't work          // What changes to call StaticMethod&lt;T&gt;()?         Sample.StaticMethod&lt;myType&gt;(); // This also doesn't work     }      public void GenericMethod&lt;T&gt;()     {         // ...     }      public static void StaticMethod&lt;T&gt;()     {         //...     } } Is an Anti-Symmetric Relation also Reflexive? According to the definition of an Anti-Symmetric Relation if xRy and yRx then x = y  Which means, effectively, x is in relation with itself. Does this mean that anti-symmetry implies reflexive property as well?