I was messing around with $$\sum_{n=2}^\infty (\zeta(n) -1)$$ and saw that it approaches 1. After putting it into wolfram, it says it does indeed converge to 1.  Like wise the sum $$\sum_{n=1}^\infty (\zeta(2n) -1) = \frac {3}{4} $$  I was wondering how these are proven? Or even if I am mistaken in assuming they're proven.
While I was looking at the values of the zeta function for the first natural numbers, I noticed that the sum of the values minus $1$, converge to $1$. Better put: $$\sum_{n=2}^{\infty} \left(\zeta(n)-1\right) = 1 $$ Furthermore, if you use only the even numbers for the zeta function, the sum will converge to $\frac{3}{4}$, or $$\sum_{n=1}^{\infty} \left(\zeta(2n)-1\right) = \frac{3}{4}$$  Leaving $$\sum_{n=2}^{\infty} \left(\zeta(2n-1)-1 \right)= \frac{1}{4}$$  This is probably common knowledge among mathematicians, but I couldn't find much about it on the internet. Is there a proof of this or perhaps even a simple explanation why this is so?
I have a lot of Excel sheets for all sorts of things - but none has ever acted up like this one here:    This used to be a totally normal Excel 2007 sheet which I was able to resize, maximize etc. - but not anymore. The MDI child window is fixed, both in position and size. I cannot move it around, more can I maximize it to use the full size of the screen anymore. I cannot double-click on it's title bar to maximize it either :-(  What happened? What can I do to "undo" whatever I did to it.  Any ideas? I tried to share / undo sharing, I tried to protect / unprotect the sheet - nothing seems to work... Difference between 'max distance' and 'radius' in Physics.SphereCastAll How can I specify a custom date formate to be validated with the  for jQuery? I am seeing quite a few conversions which have as source "accounts.google.com" (almost 40%).  AFAIK, in the new Universal Analytics as soon as a user has a Referrer set a new session is started. So if a user is getting on my page, and creates an account using social auth (e.g. Google accounts) when being redirected back, will this create a new session? I guess this would explain that there are many conversions coming from accounts.google.com - is it safe to enter accounts.google.com into the Referral Exclusion List? And more - what about Facebook logins, those would show up as facebook.com, but i probably don't want to exclude facebook.com as referrer?  Any ideas how to deal with UA referrers and social logins? Recovery of deleted folders in home folder By accident I've removed some folders (they were empty) in my home folder. I've created a new ones but they doesn't have proper icons etc... how can I recreate them step by step? cin and getline skipping input I know there are some command-line applications like pdflatex that can help me to compile a LaTex document from C#.Net (using the System.Diagnostics.Process class) .   I wanna know that what's the best command-line tool for compiling a TeX document ? Pull down resistor on logic gate input I would like to use a pull-down resistor on the input of a NOT logic gate. I am using a 5V microcontroller, and I want to ensure that the input of the gate will never be floating (eg. microcontroller damage).  General information:  The logic gate I am using is a HD74LS04 inverter whose datasheet can be found . I am testing the output by placing my positive multi-meter lead on R4 and the negative lead on GND. With the digital pin floating, the output of 1Y is 4.33V. This is the expected behavior.  The logic gate requires a minimum of 2.0V for Vih. If I understand this circuit correctly, I expect the voltage at pin 1A to be 3.76V since R2 and R3 will form a voltage divider. This value seems plenty to signify a 'high' in the logic gate.  R2 exists to protect the digital pin. Ideally, I would have liked R2 to be 1KΩ. Increasing the value of R2 means I have to increase the value of R3 to maintain the same voltage divider. During testing I found that if R3 was any value over 3.3KΩ, the pin 1A would not get pulled down all the way if I left the digital pin floating.   Tests: If I disconnect everything except GND and Vcc and place the positive lead of my multi-meter on pin 1A (now floating) and the negative lead to GND, I read 1.49V. Placing a pull-down resistor (R3) on pin 1A to GND with a value of 100Ω changes the reading to 0.02V. Increasing the resistor's value to 330Ω increases the reading to 0.07V. At the value I need, 3.3KΩ, the reading increases to 0.60V. Finally, changing the resistor's value to 1MΩ increases the reading to 1.47V and actually changes the output of pin 1Y to low.  The test that uses the 3.3KΩ resistor is concerning. The reading I took (0.60V) seems to be approaching the Vil maximum of the logic gate which is 0.80V.   Questions:  This must be a trivial question, but why does an R3 value of 3.3KΩ not pull pin 1A all the way to GND? Is there a formula for determining the correct pull-down resistor value? Other answers I found on here simply state 10KΩ as a 'good' value for a pull-up resistor. I'm not sure where that value comes from or if it is applicable to a pull-down resistor.         &ndash; Schematic created using   Edit: Although Ignacio's answer technically answers the question, I thought I would share some information I found for others who may come across this question.  User 'Papabravo' on the  states that the calculation for a pull down resistor can be calculated by taking the Vil (0.8V) and dividing it by the "standard TTL load" of 1.6mA. As Ignacio pointed out, my specific logic gate can source up to 0.4mA, so the calculation would be 0.8/0.0004A which equals 2KΩ. For safety, I performed the calculation at 0.5V in order to give myself some headroom.  Other Resources: do { ... } while (0) — what is it good for? Possible Duplicate:          I've been seeing that expression for over 10 years now. I've been trying to think what it's good for. Since I see it mostly in #defines, I assume it's good for inner scope variable declaration and for using breaks (instead of gotos.)  Is it good for anything else? Do you use it? Is it possible to configure a TP_Link ADSL router to work with a cable connection? Change style of labels of Tikz nodes of a given style I'm working with some code that widely uses the idiom of returning a pointer to a static local variable.  eg:  char* const GetString() {   static char sTest[5];   strcpy(sTest, "Test");   return sTest; }   Am I right in thinking that this is safe?  PS, I know that this would be a better way of doing the same thing:  char* const GetString() {   return "Test"; }   Edit: Apologies, the function signature should of course be:  const char* GetString(); How handicapped am I in graduate admissions if I graduated from a lower tier university? Due to a combination of factors, I ended up going for my undergraduate studies to a relatively unknown university that was close to me geographically.  Let's say that I have a relatively standard background for someone trying to apply to top tier schools for graduate studies (some research experience, some low to medium quality publications, very strong grades, good recommendation letters). Then how much does the alma mater have an impact for someone like me? I obviously know that it is still possible to be accepted, but how much of a disadvantage am I looking at exactly? I work in D8, I create a custom module first_module. By modules create a url and display all nodes by *.html.twig file.  Write code in controller.php  public function contentthree() {    $connection = Database::getConnection();    $query = $connection-&gt;select('node', 'n');   $query-&gt;fields('n', ['nid', 'type']);    $groups = $query-&gt;execute()-&gt;fetchAll();    foreach ($groups as $group) {     $events[] = [$group-&gt;nid, $group-&gt;type];   }    return [     '#theme'  =&gt; 'first_module_display',     '#events' =&gt; $events,   ]; }   In *.module  function first_module_theme($existing, $type, $theme, $path) {    return [     'first_module_display' =&gt; [       'variables' =&gt; ['events' =&gt; NULL],       'template'  =&gt; 'first-module-list',     ],   ]; }  function _get_node_details() {    $output = 'Welcome D8';    $output .= 'Node Details';    return $output; }   and create templates first-module-list.html.twig and display data  {%for key, data in events %}  {%for key1, data1 in data %}     {{ data1 }}  {% endfor %}  {% endfor %}   work fine and showing data correct. But, my question is how to call function _get_node_details() and how to render node details in first-module-list.html.twig files? What is the yield keyword used for in C#? In the  question one of the answers had the following code snippet:  IEnumerable&lt;object&gt; FilteredList() {     foreach(object item in FullList)     {         if(IsItemInPartialList(item))             yield return item;     } }   What does the yield keyword do there?  I've seen it referenced in a couple places, and one other question, but I haven't quite figured out what it actually does.  I'm used to thinking of yield in the sense of one thread yielding to another, but that doesn't seem relevant here. Display only the penultimate (second last) row of a text How can I distinguish between words which have the -ing in a sentence that are nouns, verbs, or adjectives?    For example sometimes -ing come with word to give us a noun, and sometimes a verb or an adjective.       That interesting snake.   That crawling snake.   That barking dog keeps everyone awake.     Is there any rule to determine whether a word with -ing is a noun, verb, or adjective? Recall the following relevant definitions. We say that   $b$ is divisible by $a$ in $R$, or $a\mid b$ in $R$, if $b = r a$ for some $r\in R$. $a$ and $b$ are associates  in $R$ if $a\mid b$ and $b\mid a$, (or, equivalently, if $aR = bR$). $u\in R$ is a unit if it has a multiplicative inverse (a $v\in R$ such that $uv=vu=1$). $a$ and $b$ are unit multiples in $R$  if $a = ub$ for some unit $u\in R$.   Given these definitions, my question is,     If $R$ is a commutative ring with unity and $a,b\in R$ are associates in $R$, are $a$ and $b$ unit multiples in $R$?   I was told that this not always true. But I encountered some difficulties in finding a counterexample. Calculation of subgroups of $(Z_{12}, +)$ How calculate all subgroups of $(Z_{12}, +)$?  I know that the order of subgroups divide the order of the group, but there is such a smart way to calculate the subgroups of order 6? List of all "partitions"/ "cuttings" of a list