Let $\frac{1}{2} + 1 + \frac{9}{8}+1+\frac{25}{32}+\frac{36}{64}+\cdots$ be a series. I need to calculate its sum using $$\sum_{n=1}^{\infty}n^2x^n.$$ I was trying to set $x=\frac{1}{2}$, so: $$\sum_{n=1}^\infty n^2x^n=x\sum_{n=1}^\infty n^2x^{n-1}=x\sum_{n=1}^\infty (nx^n)'=x^2\sum_{n=1}^\infty (x^n)''.$$ Then I was differentiating separately: $$x^2\sum_{n=1}^\infty (x^n)'' = x^2\left(\frac{1}{1-x}\right)''=x^2\left(\frac{1}{(1-x)^2}\right)'=x^2\frac{2(1-x)}{(1-x)^4}=\frac{2x^2}{(1-x)^3}.$$ And finally calculate the sum: $$\frac{2 \cdot \frac{1}{4}}{\frac{1}{2}^3} = 4.$$ But I got the wrong answer. What did I miss? How do I solve it correctly?
It seems that I'm missing something about this.  First of all, the series is convergent: $\lim_{n\rightarrow\infty}\frac{2^{-n-1} (n+1)^2}{2^{-n} n^2}=\frac{1}{2}$ (ratio test)  What I tried to do is to find a limit of a partial sum $\lim_{n\rightarrow\infty}S_n$ as follows: $S_n=\frac{\frac{1}{6} n (n+1) (2 n+1)}{\frac{1-\left(\frac{1}{2}\right)^n}{2 \left(1-\frac{1}{2}\right)}}$. Still, the limit is $\infty$ and I'm clearly doing something wrong.
The English alphabet has a common pronunciation. For example, the letter b is pronounced like the word bee, the letter c like the word see, and the letter i like the word aye.  Is there a formal spelling for the letter names? I have an ASUS E402S laptop with Ubuntu 64-bit 16.04 LTS installed. The main problem is that the OS does recognize the touchpad with basic functions, but additional options such as scrolling through pages or between open applications are non-existent.  This is what my xinput result is showing:  ⎡ Virtual core pointer                      id=2    [master pointer  (3)] ⎜   ↳ Virtual core XTEST pointer                id=4    [slave  pointer  (2)] ⎜   ↳ Logitech USB Receiver                     id=9    [slave  pointer  (2)] ⎜   ↳ Logitech USB Receiver                     id=10   [slave  pointer  (2)] ⎜   ↳ FTE1000:00 0B05:0101                      id=12   [slave  pointer  (2)] ⎣ Virtual core keyboard                     id=3    [master keyboard (2)]     ↳ Virtual core XTEST keyboard               id=5    [slave  keyboard (3)]     ↳ Power Button                              id=6    [slave  keyboard (3)]     ↳ Video Bus                                 id=7    [slave  keyboard (3)]     ↳ Sleep Button                              id=8    [slave  keyboard (3)]     ↳ USB2.0 VGA UVC WebCam                     id=11   [slave  keyboard (3)]     ↳ Asus WMI hotkeys                          id=13   [slave  keyboard (3)]     ↳ AT Translated Set 2 keyboard              id=14   [slave  keyboard (3)]   Apparently my touchpad's type is not shown in the list. What are the possible solutions to fix this issue? Black is Back! New update with Achievement shows black backgrounds with black text Update: This is fixed on version 1.0.43 as per the accepted answer below.  Already saw ,  and . They are old and seem to be about previous versions.  I just updated to the version 1.0.41 of the Android app that brings the new feature - Achievements. Sadly, I think the stress of World Cup has made you guys not do a good job with it. It's buggy.  When I open my Achievements panel and tap on any post, I get this:    After I get this, it persists throughout the rest of the app. For example, if I go to the feed and open another question, I get this too.  Detailed reproduction steps are available in AD7six's answer.  My Device: Samsung Galaxy SIII LTE GT-I9305 Android: Stock ROM 4.3 Jelly Bean  Other Devices: - Device model, Android version, Reported by - HTC Desire, 4.2.2, xd6_ - Huawei Ascend, 4.2, Manu343726 - LG Optimus F3, 4.1.2, Rocket Hazmat - LG Optimus G, 4.3, Canadian Luke - Samsung Galaxy Nexus (Sprint), 4.3, Chrono Kitsune - Samsung Galaxy Note II, 4.3, Katja Christiansen - Samsung Galaxy SII GT-I9100, 4.1.2, AD7six - Sharp SH837W, 4.1.2, timrau - Sony Xperia SP, 4.3, pepoluan - Sony Xperia Z1, 4.3, user2313067 - Unknown, 4.2.1, Deer Hunter What settings should I be using to shoot "Supermoon"? In "", Mark Lutz mentions "mixins". I'm from a C/C++/C# background and I have not heard the term before. What is a mixin?   Reading between the lines of  (which I've linked to because it's quite long), I'm presuming it's a case of using multiple inheritance to extend a class as opposed to 'proper' subclassing. Is this right?   Why would I want to do that rather than put the new functionality into a subclass? For that matter, why would a mixin/multiple inheritance approach be better than using composition?  What separates a mixin from multiple inheritance? Is it just a matter of semantics? Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! How can I prepend a tag to the beginning of several files? I have to try to add a field called 'Name' to a bunch of different feature classes using python, and then access the name of the feature class to fill in the field. I have no idea how to even start this! I know I need a loop but that's all I know. Conveying "under the radar" in a less colloquial way My sisters and I all have iPhones under the same account. I also have an iPad under the same account. I am trying to get the iMessage on my iPad compatible with my phone number, not email address.   The only phone number choice to choose from is my sister's iPhone number. How can I choose my phone number for iMessage on my iPad? example:   Does Chocolatey run the native program installer? 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