How to Find $$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$$ the most basic way possible?   I know it's $\pi^4/90$ but to arrive at this figure? Curious, because I need it to solve the integral $\int_{0}^{\infty}x^3\cdot(e^x-1)^{-1}\;dx$.
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$.  For example, see  or the answers to the question "?"  Are there any nice ways to prove that $$\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}?$$  I already know some proofs that give all values of $\zeta(n)$ for positive even integers $n$ (like #7 on Robin Chapman's list or Qiaochu Yuan's answer in the linked question).  I'm not so much interested in those kinds of proofs as I am those that are specifically for $\zeta(4)$.  I would be particularly interested in a proof that isn't an adaption of one that $\zeta(2) = \pi^2/6$.
What is Renly's basis for removing Joffrey? Related question:   I get that he considers himself a better king than Stannis, but why isn't he ?  Did Renly know about Joffrey being illegitimate? I always thought he didn't and was distrustful that Cersei would accept Ned as Regent.  I think in both the books and the series, Renly finds out Joffrey is illegitimate from Stannis' letters, but rewatching that clips makes me think Renly was with Ned, Varys and LF in the plan to arrest Joffrey and Cersei.  [looks up something]  Oh also . So Renly did know? Whenever a question pops up on SO about Java synchronization, some people are very eager to point out that synchronized(this) should be avoided. Instead, they claim, a lock on a private reference is to be preferred.  Some of the given reasons are:    (very popular this one, also has an "accidentally" variant)     Other people, including me, argue that synchronized(this) is an idiom that is used a lot (also in Java libraries), is safe and well understood. It should not be avoided because you have a bug and you don't have a clue of what is going on in your multithreaded program. In other words: if it is applicable, then use it.  I am interested in seeing some real-world examples (no foobar stuff) where avoiding a lock on this is preferable when synchronized(this) would also do the job.  Therefore: should you always avoid synchronized(this) and replace it with a lock on a private reference?      Some further info (updated as answers are given):   we are talking about instance synchronization both implicit (synchronized methods) and explicit form of synchronized(this) are considered if you quote Bloch or other authorities on the subject, don't leave out the parts you don't like (e.g. Effective Java, item on Thread Safety: Typically it is the lock on the instance itself, but there are exceptions.) if you need granularity in your locking other than synchronized(this) provides, then synchronized(this) is not applicable so that's not the issue "Dummy Output" No sound in Ubuntu 16.04 None of the HDMI output are working, all saying "dummy output", it was working yesterday, but when I use the headphones it became soundless, with or without the headphones.  Thank you already! Do I have to use “do” in any “wh-” question? What did you say?  What you said?   What did you wear? What you wore?    Where did you go? Where you go?    I know that sometimes in English, people do not use “do” in “wh-” questions. I have heard somewhere that people do not use “do” when the answer is a proper noun or a unique thing. Am I right? In the LaTeX document pasted below, I still see page numbers upon compiling, even though I've included a \pagestyle{empty} command below the \maketitle command. Any ideas what I'm doing wrong? Thanks in advance.  \title{Thesis Abstracts} \author{         Kurt Han Peek } \date{\today}  \documentclass[a4paper,11pt]{article} \usepackage{graphicx} \usepackage[margin=2cm]{geometry}  \begin{document} \maketitle \pagestyle{empty}  \section*{An Analysis of the Effects of Digital Phase Errors on the Performance of a FMCW-Doppler Radar} In modern frequency-modulated continuous-wave (FMCW) radars, the transmitter is increasingly being implemented using direct digital chirp synthesis (DDCS), which provides improvements in sweep linearity, stability, precision, agility, and versatility over analog techniques. Its main limitations are errors due to sampling of the modulating signal, phase truncation, and digital-to-analog converter (DAC) quantization, which produce spurious signals due to their deterministic nature. This thesis presents an analysis and simulation of the effect of two sources of digital phase errors -- namely, sampling of the modulating signal and phase truncation -- on the performance of a FMCW-Doppler transceiver employing a DDCS in its transmitter. An upper bound for the amplitude of spurious targets resulting from these digital phase errors is established. Further, it is shown that provided the phase errors are periodic with the sweep period, the spurious targets are not offset from the target in Doppler. An algorithm for selecting the digital chirp parameters of a DDCS so as to ensure periodic and phase-continuous sweep transitions, which guarantee periodic phase errors and reduce `spectral splatter', is devised. Finally, we investigate parallels of FMCW radar in the optical domain, and consider the fundamental question whether range resolution is fundamentally limited by the bandwidth of a transmitted signal, or its carrier frequency.  \section*{Estimation and Compensation of Frequency Sweep Nonlinearity in FMCW radar} One of the main issues limiting the range resolution of linear frequency-modulated continuous-wave (FMCW) radars is nonlinearity of frequency sweep, which results in degradation of contrast and range resolution, especially at long ranges. Two novel, slightly different, methods to correct for frequency sweep nonlinearity by digital post-processing of the deramped signal were introduced independently by Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003) and Meta et al. (Meta, Hoogeboom et al. 2006). In these publications, however, no formal proof of the techniques was given, and no limitations were described. In this thesis, we prove that the algorithm of Meta is exact for temporally infinite chirps, and remains valid for finite chirps with time-bandwidth products which are large compared to unity, provided the maximum frequency component of the phase error function is small compared to the square root of the nominal chirp rate. It is also shown that the algorithm of Meta reduces to that of Burgos-Garcia in this limit. A digital implementation of both methods described. Finally, we propose a novel method to measure the systematic phase errors which are required as input to the compensation algorithm.  \end{document} Inspired by a , here's a list of the 'etymology' for the  hats. For pictures and a list of criteria, see  instead.  The list is a community wiki; feel free to edit it if you have more information about a hat. If you like to do some research, most hats have been featured in earlier editions, so you might be able to find more information in old Meta Stack Exchange questions. Is a 1st-level spell cast using a 4th-level slot a 1st-level spell, or a 4th-level spell? As the title says: Is a 1st-level spell cast using a 4th-level slot a 1st-level spell, or a 4th-level spell? Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it should make no difference.)  Why $(f * g) (x) = \int f(y) g(x - y) dy$ rather than $(f * g) (x) = \int f(y) g(x + y) dy$?  Edit: I'm finding it really hard to choose a best answer. There are at least three very good ones here. $X$ and $Y$ are independent $U(0,1)$ random variables.   I have to find $E[X|X&gt;Y]$.   I found it out by integrating $\int_0^1 \int_y^1 x dxdy$ to get $\frac{1}{3}$ as the answer but this is wrong according to the answer key. What is the mistake that I have made? Emails Per Invocation Still Capped at 100? Microsoft Edge keeps taking over PDF file open/association Ubuntu installation is slow, why? Unexpected output for this following code Is it okay to copy-paste answers from other questions? This one leaves me with a bad taste in my mouth, but I defer to others as to whether it is appropriate or not. In , I wrote an answer to a question based on my experiences; it was upvoted and accepted, woo.  Now, today, in response to a different &#8212; but related &#8212; , someone has pasted my answer nearly verbatim (awkward English intact, I may add) as an answer.  Is that really okay? Is it a behavior that should be encouraged? Discouraged? Taylor's polynomial uniqueness proof - why are these limits inferable? So I'm viewing a short proof on the uniqueness of Taylor polynomials.     Uniqueness of Taylor polynomial:   Let $f:]a,b,[ \rightarrow \mathbb{R}$ $n$ times continuously   differentiable and $x_0 \in ]a,b,[$. If $p$ is $n$th degree polynomial   function for which       $$f(x)-p(x)=o(|x-x_0|^n), \space \text{when} \space x \rightarrow x_0$$  then      $$p(x)=T_{n, x_0} f(x)$$   Proof starts like:  $$\frac{p(x)-T_{n, x_0}f(x)}{(x-x_0)^n} = \frac{p(x)-f(x)}{(x-x_0)^n}+\frac{f(x)-T_{n, x_0}f(x)}{(x-x_0)^n}$$  then   $$\frac{p(x)-f(x)}{(x-x_0)^n} \rightarrow 0 \space, when \space x-x_0 \rightarrow 0$$  $$\frac{f(x)-T_{n, x_0}f(x)}{(x-x_0)^n} \rightarrow 0 \space, when \space x-x_0 \rightarrow 0$$  Now my problem is, how can one actually say that the above approach $0$ if the denominator approaches $0$? After T-2, why not send back another T-800? This might be a completely wrong question, but this is bothering me since many days ago. Given the mass (Sun) curves the space around it, gravitation is the result of such curved space (Correct me if I am wrong, source: A Documentary film). Given any point on a circle with center same as center of the mass, curvature in the space should be equal (Intuition).   Planets rotate around the Sun because of the curve in the space they should follow a circular path and the distance between planet and Sun should be at a distance.  Given the fact that earth has a elliptical orbit around the sun, and the distance between Earth and Sun varies according to position of the earth. Why do we have a elliptical orbit not a circular orbit. Including pdf_tex images from inkscape How to convert every pixel of raster picture to square objects in Illustrator? As of now, my desktop has begun showing the contents of my home folder (~) instead of the contents of ~/desktop. I must have changed something, but I have no idea what, or what I was doing exactly when it happened.  How do I re-point gnome to the right folder?  I'm running Ubuntu 11.04.