I have been solving problems using a "Potentially Helpful Formulas" sheet from my esteemed math professor. i want to solve for:   $\sum_{n=1}^{\infty} \dfrac{1}{n^4} =$ ?  On my formula sheet i have:  $\sum_{n=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}$  Which looks similar to what i want, but not quite. Might someone guide me towards the solution (or provide the solution with explanation?
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 $\zeta(2) = \pi^2/6$.
I have the following browsers on my phone:   stock Dolphin Mini Opera Mobile Firefox   I don't want to set a catch-all default because some pages render better on some browsers than on others.  When I click on a link on a non-browser app (e.g. google search or google reader) it asks me which browser to choose (as expected).  Can I set a default browser for some pages and a different default for others? How long would the conference be for? I want to know if this is grammatically correct " How long would the conference be for?". I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no idea how induction worked, I showed him some typical examples.  Now he showed me an exercise he was having trouble with, which states that one should prove $$n^{n+1} &gt; (n+1)^n$$  for all $n \geq 3$. I have to admit that I too have trouble showing this inequality, as in all of my attempts, my lower bound is too low. Also, I have not yet figured out, how one gets the $(n+2)^{n+1}$, primarely the number $2$ is a problem. I think this might be solved using the binomial theorem, however, I don't think they have already seen the binomial theorem in school.  Is there an easy method to show this inequality by induction not using the binomial theorem? If no: How can one show it using the binomial theorem?  Thanks for answers in advance. Polynomial degree 3 Equation Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$. I'm reading Intro to Topology from Mendelson.  The entire problem statement is,  Prove that $X$ is compact if and only if for each family $\lbrace F_\alpha\rbrace_{\alpha\in I}$ of closed subsets of $X$ that has the finite intersection property, we have $\bigcap_{\alpha\in I} F_\alpha\neq\varnothing$.  My attempt at the proof is as follows:  First assume that $X$ is compact. For the sake of contradiction, suppose that there exists a family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with FIP such that $\bigcap_{\alpha\in I} F_\alpha=\varnothing$. Since $X$ is compact if $\lbrace F_\alpha\rbrace_{\alpha\in I}$ any family of closed sets such that $\bigcap_{\alpha\in I} F_\alpha=\varnothing$, then there exists a finite set of indices $\lbrace \alpha_1,\dots,\alpha_n\rbrace$ such that $\bigcap\limits_{i=1}^n F_{\alpha_i}=\varnothing$. Yet, since $\lbrace F_\alpha\rbrace_{\alpha\in I}$ has the finite intersection property, for every finite index of $J\subset I$, $\bigcap\limits_{\alpha\in I} F_\alpha\neq\varnothing$.  Suppose now that $X$ is not compact, that is, there exists an open cover $\lbrace U_\alpha\rbrace_{\alpha\in I}$ of $X$ with no finite subcover. Consider the set $F_\alpha=C(U_\alpha).$ Then $\lbrace F_\alpha\rbrace_{\alpha\in I}$ is a collection of closed subsets of $X$ with the finite intersection property and were $\bigcap_{\alpha\in I} F_\alpha=\varnothing.$  Thanks for any feedback! Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$ This is a homework question whereby I am supposed to evaluate:  $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$  Wolfram Alpha outputs the answer as   $$\frac{1}{2}(\pi \coth(\pi) - 1)$$  But I have no idea how to get there. Tried partial fractions (by splitting into imaginary components), tried comparing with the Basel problem (turns out there's little similarities), nothing worked. (p.236)  has a proof that the equation $e^z-z=0$ has infinitely many complex solutions:    I'm curious if there are any particularly elegant ways to see this, other than that given in the text. Unable to SSH to my virtual machine via Windows Power Shell ISE A get() like method for checking for Python attributes If I had a dictionary dict and I wanted to check for dict['key'] I could either do so in a try block (bleh!) or use the get() method, with False as a default value.  I'd like to do the same thing for object.attribute. That is, I already have object to return False if it hasn't been set, but then that gives me errors like     AttributeError: 'bool' object has no attribute 'attribute' Is it possible to install Ubuntu without formatting any partitions? What is the Stack Exchange (Staff) process of launching a site? What is the difference between "till" and "until"? What is the difference between till and until?    When to use till or until? Please explain with examples. How to decide if an effect is magical for Magic Resistance? Some creatures have Magic Resistance:     The [...] has advantage on saving throws against spells and other magical effects.   We know from  that this works against Turn Undead, but does it work against Stunning Strike, Death Strike, Fey Presence, etc? How do we decide? C++ templates that accept only certain types Which Standard Visitor visa should I apply for? How to convert HTML to XHTML? I need to convert HTML documents into valid XML, preferably XHTML. What's the best way to do this? Does anybody know a toolkit/library/sample/...whatever that helps me to get that task done?  To be a bit more clear here, my application has to do the conversion automatically at runtime. I don't look for a tool that helps me to move some pages to XHTML manually. Weird syslog order Have you ever seen wrong order in /var/log/syslog ? I think that it causes only crontab.  Jun 28 22:20:01 alex CRON[2327]: (root) CMD (something &gt; /dev/null) Jun 28 22:21:01 alex CRON[2700]: (root) CMD (something &gt; /dev/null) Jun 21 03:49:01 alex CRON[2753]: (root) CMD (something &gt; /dev/null) Jun 28 22:23:01 alex CRON[2754]: (root) CMD (something &gt; /dev/null) Jun 27 05:08:01 alex CRON[2813]: (root) CMD (something &gt; /dev/null) Jun 26 00:19:01 alex CRON[2816]: (root) CMD (something &gt; /dev/null) Jun 28 22:25:01 alex CRON[2816]: (munin) CMD (if [ -x /usr/bin/munin-cron ]; then /usr/bin/munin-cron; fi) Jun 28 22:25:01 alex CRON[2821]: (root) CMD (something &gt; /dev/null) Jun 28 22:27:01 alex CRON[3225]: (root) CMD (something &gt; /dev/null) Jun 28 22:28:01 alex CRON[3240]: (root) CMD (something &gt; /dev/null) Jun 28 22:30:01 alex CRON[3290]: (root) CMD (something &gt; /dev/null) Jun 26 12:03:01 alex CRON[3291]: message repeated 2 times: [ (root) CMD (something &gt; /dev/null)] Jun 28 22:30:01 alex CRON[3291]: (munin) CMD (if [ -x /usr/bin/munin-cron ]; then /usr/bin/munin-cron; fi) Jun 28 22:31:01 alex CRON[3659]: (root) CMD (something &gt; /dev/null) Jun 28 22:32:01 alex CRON[3754]: (root) CMD (something &gt; /dev/null) Jun 16 06:41:01 alex CRON[3846]: (root) CMD (something &gt; /dev/null) Jun 28 22:35:58 alex cron[909]: (CRON) INFO (pidfile fd = 3) Jun 28 22:35:58 alex cron[944]: (CRON) STARTUP (fork ok) Jun 28 22:35:58 alex cron[944]: (CRON) INFO (Running @reboot jobs) Jun 28 22:36:01 alex CRON[1428]: (root) CMD (something &gt; /dev/null) Jun 28 22:37:01 alex CRON[3050]: (root) CMD (something &gt; /dev/null) Jun 28 22:38:01 alex CRON[3082]: (root) CMD (something &gt; /dev/null) Jun 28 22:39:01 alex CRON[3114]: (root) CMD (something &gt; /dev/null)   As you can see, there is bad time order between lines. It is just crontab error, or there is some problem with system clock, or something?  I am running the latest Ubuntu 14.04 kernel:  Linux alex 3.13.0-30-generic #54-Ubuntu SMP Mon Jun 9 22:45:01 UTC 2014 x86_64 x86_64 x86_64 GNU/Linux What to do with bounty on question which is no longer relevant? So I can start from len(collection) and end in collection[0].  I also want to be able to access the loop index. What's the intuition behind "stiff systems"?