It is ok to use the maclaurin series for $\frac{1}{x+1} $ to expand the function $$f(x) = \frac{x}{x^4+x^2+1}$$ like that:  $$f(x) = x\frac{1}{(x^4+x^2)+1} = x\sum_{n=0}(-1)^n(x^4+x^2)^n = \sum_{n=0}(-1)^nx(x^4+x^2)^n $$  $$=\sum_{n=0}(-1)^nx^{2n+1} (x^2+1)^n$$
Need fast solution how to show series expansion of function at x=0:  $$f(x)=\frac{3x}{1+x^2+x^4}$$  Please some hints.
Virtualbox Kernel driver not installed Proving that the tangent to a convex function is always below the function GCD in a PID persists in extension domains What is "superuser" and how do I get that privilege? How many ways can $r$ nonconsecutive integers be chosen from the first $n$ integers? Alright, I have been doing the following (variable names have been changed):   FileInputStream fis = null; try {     fis = new FileInputStream(file);      ... process ...  } catch (IOException e) {     ... handle error ... } finally {     if (fis != null)         fis.close(); }    Recently, I started using FindBugs, which suggests that I am not properly closing streams.  I decide to see if there's anything that can be done with a finally{} block, and then I see, oh yeah, close() can throw IOException.  What are people supposed to do here?  The Java libraries throw too many checked exceptions. Stack Exchange iOS app crashes on start Upon clicking on SE icon, blue (starting) screen appears and then I am back to home screen, SE app simply won't start. This kind of behaviour never happened before (on any other app). I tried to remove it from &quot;task manager&quot;, reinstalling and logging back in; neither did work. In Settings -&gt; Stack Exchange -&gt; Notices I found a log message:  FILE DOESN'T EXIST, WILL CREATE: users/dev/buildagent/work/{id}/pods/target support files/pods/pods-ack...plist   App Version: 1.4.0 Device: iPhone7,2 OS Version: Version 9.1 (Build 13B143)   TLDR: app starts when cellphone has no internet access. I just noticed when I am on 3G (cellular internet) it crashes ASAP, but when I am on wifi that has no internet (for LAN etc.) app starts normally but no new content, when I turn off WiFi and iPhone switches to cellular; app crashes. Note: I turn off WiFi using control center - so I never leave the app. Another thing, when I force iPhone to use EDGE (slowest possible internet) app starts, and crashes after 5 seconds or so. I assume when app downloads some data from server.  Known workaround (works for me too)  I found a workaround: turn on Airplane Mode, launch the app, then turn Airplane Mode off. – NobodyNada How can I share files over a Windows network? I have a media computer which I recently reformatted to Ubuntu 10.04. I am probably just missing something obvious but I can't figure out how to share folders and files, so that I can browse to them on my Windows computers in order to copy files to/from the computer.  How do I do this? indentation of a whole paragraph Minecraft extra folders? Once I click to run minecraft a small window appears saying "Downloading runtime", then three other folders appear onto my desktop: Game, Native Log, and Runtime. This really annoys me, as I have to keep deleting them after I am done with a session of minecraft. Why is this? It never use to happen before. Tags on badge question list I'm trying to get the Weed-Eater hat on Stack Overflow. To do so, I'm looking at the list of questions . The problem is, no tags show up there, and so it's rather difficult to see which ones I have a shot at. Any chance of showing the tags on this page?  I realize that getting this before Winter Bash is over would be difficult, but it would be good to keep such things in mind for the future. I have the value 25.00 in a float, but when I print it on screen it is 25.0000000. How can I display the value with only two decimal places? Make flash drive bootable with MULTIPLE Windows Installers How to subtract days from a plain Date? Split List into Sublists with LINQ Is there any way I can separate a List&lt;SomeObject&gt; into several separate lists of SomeObject, using the item index as the delimiter of each split?  Let me exemplify:  I have a List&lt;SomeObject&gt; and I need a List&lt;List&lt;SomeObject&gt;&gt; or List&lt;SomeObject&gt;[], so that each of these resulting lists will contain a group of 3 items of the original list (sequentially).  eg.:   Original List: [a, g, e, w, p, s, q, f, x, y, i, m, c] Resulting lists: [a, g, e], [w, p, s], [q, f, x], [y, i, m], [c]   I'd also need the resulting lists size to be a parameter of this function. LED light fixture glows when not turned on This past week at Menards I bought 2 Smart Electrician® 3650 Lumens 46" LED Linkable Shop Light Model Number: 0053-0011  |  Menards® SKU: 3481400.  The LED shop light fixture has a glow to it. Looked like the LEDs were turned on, but not all the way. The power switch is turned off. Don’t know why I did this but I took one of those three prong adaptors converters to two prong and to my surprise the light fixture didn’t light up. I unplug the cord with the adaptor and turned it around and it still didn’t light up. The power supply electronics inside the light fixture must be using the earth ground wire somehow. Using my volt meter. I measured to earth ground to the green wire that comes out of the adaptor and to my surprise depending how I plug in the adaptor. I’m getting 112 volts AC. That's because the adaptor isn't keyed and I'm switching the HOT and NEUTRAL around depending how I  plug it in.  I connected the LED light fixture to “Safety Analyzer tester”. When I press the “Open Ground” button, the fixture stops glowing (lightly lite LED’s).  Looks like the power supply circuit inside the fixture has a connection to “Earth Ground”.  I then switched the “Normal” on the polarity switch to “Reverse” and the LED’s were off. Switching the Neutral to open position has no effect. I'm thinking of taking them apart and see inside what's going on or take them back.  I figured out the problem.    In this light fixture you can clearly see the white wire (neutral/common) is connected to the light switch, turning the power off the LED light strip. Commonly we switch off the “HOT” to your light socket so when you change your bulb you have a less of a chance of getting electrocuted (that’s another story for later). Sometimes in Power supply circuits they place capacitors in there to help filter electrical noise out. Also known as suppressers (electronics to reduce unwanted noise or oscillation in a circuit or unwanted frequencies in a signal). Depending on how or where they are placed in the circuit it could possible cause some issue if not installed correctly or if the circuit isn’t designed correctly.   Since the BLACK live wire is connected to the LED PCB all the time, anything connected to that line will potentially be LIVE voltage, that way when I disconnected the earth ground from the socket and measured 112 volts on it. My guess is they have capacitors connected to both LIVE and NEUTRAL to the earth ground wire.  The current is flowing through capacitor C2, just enough to turn the LED’s on slightly. If the switch was on the hot side, this problem wouldn’t be happening. I still may use them just not connect the “earth ground” wire. Because that’s providing the return path for lighting the LED’s. Does $\exp(2ir\pi)$ equal $1$? What's wrong? Since $e^{ix}=\cos x+i\sin x$, thus  $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$     Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$    But this cannot be true since $\cos2\pi r+i\sin2\pi r$  isn't necessarily $1$ for arbitrary $r$. Where did it go wrong? How do I make exponential equation with points?  Here are my points:  x = 0, y = 0 x = 1, y = 4.75000 x = 2, y = 9.26250 x = 3, y = 13.54937 x = 4, y = 17.62191   this is same as   x = 0, y = 0 x = 1, y = (0 + 5) * 0.95  x = 2, y = ((5 * 0.95) + 5) * 0.95 x = 3, y = (((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 x = 4, y = ((((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 + 5) ) 0.95   as x value increase by 1, y value is added by 5 and multiplied by 0.95 i tried to make an equation for this but i couldn't  also i tried to use natural exponential but the initial value is 0 so it didn't work  please help me What does string theory predict for the singularity inside a black hole? The usual explanation for what's going on inside a black hole goes something like "General Relativity predicts a singularity with infinite curvature, but when matters gets so tightly compressed we should expect quantum effects to become important, so the singularity will probably be blurred". Authors then proceed to say that we need a quantum theory of gravity to answer such things, and quickly abandon the subject.  Well, while we don't have an experimentally confirmed quantum theory of gravity, we do have a speculative one, string theory. What does it say about singularities? Are they smeared out like the books say? Do they remain points (or rings for a rotating black hole) with infinite curvature and density? Do we not know? Changing Custom Post Type Name Order