Question asks find the sum of the series.  Firstly i tried to use geometric series formula to calculate sum but could not convert it to geo form. $$ ar^{n-1} $$ So should we transform equation into geo form or use another method to calculate sum?  This is the equation i try to calculate sum.  $$\sum_{n=0}^∞ = \frac{1}{(n+1)2^n}$$
What is the sum of $$\sum_{n=0}^{\infty}\frac{1}{2^n(n+1)}$$    I've spent an insane amount of time on this problem. I checked on Wolfram and it gives $ln(4)$, which I assume you get from $2\ln(2)$.   I know that the integral of $\frac{1}{n+1}$ is $\ln(n+1)$. What am I missing?  Thank you!
Where are the evolutionary "inbetweeners"? The problem I've always had with evolution is the actual lack of variation between animals. More specifically, the lack of observable gradual change between species.  Take for example the hammerhead shark. It obviously has a strong connection with a normal shark, but where's the "half-hammerhead" shark? Where's the species that only has a SLIGHTLY wider head?  I really doubt that a single mutation on a normal shark resulted in a head shaped like that, it had to happen gradually. Right? As far as I understand, evolution works by random mutation in a child being somehow slightly beneficial to that animal so it get's to live longer and reproduce more, eventually resulting in it's genetic material in dominating the inferior ones before it. But if it was for whatever reason beneficial to have a slightly wider head, why don't all the sharks have slightly wider heads now? On the other hand, if it wasn't beneficial, how come the hammerhead still exists?  Furthermore, I feel like some characteristics of a certain animal can't really have gradual benefits. There's a lizard that shoots blood out of it's eyes. The original ancestor certainly didn't, and I'm sure that if it's child only bled slightly from it's eyes it wouldn't really help if much in life. So for it to actually become beneficial, it seems to me that there had to be a ton of lizards that had a disadvantage of bleeding slightly. In fact, there had to be so many that they evolved even further in the same direction and kept doing that until they bled so much it's an actual benefit (sorry for this turning so gross). So why did the inbetweeners survive long enough to evolve even more, and why did they THEN die out? If $A=\left( \begin{array}{ccc} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \end{array} \right)$, find the rank of $A+A^2+A^3$  We have $\rho(A+A^2+A^3)\leq\rho(A)+\rho(A^2)+\rho(A^3) $ where $\rho(A)$ means rank of A.  Here $\rho(A)=3.$ Also $\rho(A^2)=3$ and $\rho(A^3)=3$  as $\det A\neq 0$ and $\det(A^2)=\det(A)\det(A)\neq 0$ and so on.   But how to get the desired result. Using \vfill between two texts works:  \newpage bla \vfill blabla   but it doesn't work if there is no text before it:  \newpage \vfill blabla   How can this be achieved? An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Second derivative "formula derivation" I've been trying to understand how the second order derivative "formula" works:  $$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$  So, the rate of change of the rate of change for an arbitrary continuous function. It basically feels right, since it samples "the after $x+h$ and the before $x-h$" and the $h^2$ is there (due to the expected /h/h -> /h*h), but I'm having trouble finding the equation on my own.  It's is basically a derivative of a derivative, right? Newtonian notation declares as $f''$ and Leibniz's as $\frac{\partial^2{y}}{\partial{x}^2}$ which dissolves into:  $$(f')'$$ and $$\frac{\partial{}}{\partial{x}}\frac{\partial{f}}{\partial{x}}$$  So, first derivation shows the rate of change of a function's value relative to input. The second derivative shows the rate of change of the actual rate of change, suggesting information relating to how frequenly it changes.  The original one is rather straightforward:  $$\frac{\Delta y}{\Delta x} = \lim_{h\to0} \frac{f(x+h) - f(x)}{x + h - x} = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$  And can easily be shown that $f'(x) = nx^{n-1} + \dots$ is correct for the more forthcoming of polynomial functions. So, my logic suggests that to get the derivative of a derivative, one only needs to send the derivative function as input to finding the new derivative. I'll drop the $\lim_{h\to0}$ for simplicity:  $$f'(x) = \frac{f(x+h) - f(x)}{h}$$  So, the derivative of the derivative should be:  $$f''(x) = \lim_{h\to0} \frac{f'(x+h) - f'(x)}{h}$$  $$f''(x) = \lim_{h\to0} \frac{  \frac{ f(x+2h) - f(x+h)}{h} - \frac{ f(x+h) - f(x)}{h}  }{h}$$  $$f''(x) = \lim_{h\to0} \frac{  \frac{ f(x+2h) - f(x+h) - f(x+h) + f(x)}{h}  }{h}$$  $$f''(x) = \lim_{h\to0} \frac{  f(x+2h) - f(x+h) - f(x+h) + f(x)  }{h^2}$$  $$f''(x) = \lim_{h\to0} \frac{  f(x+2h) - 2f(x+h) + f(x)  }{h^2}$$  What am I doing wrong? Perhaps it is the mess of it all, but I just can't see it. Please help. In a table of contents that has been generated with LaTeX's \tableofcontents command, I would like to add a vertical space before a Chapter item in the table of contents. One way that I have in my notes is to use  \addtocontents{toc}{\protect\vspace{10pt}}   at a desired location in the document so that this gets processed by \tableofcontents to include an appropriate vertical space in the table of contents. I have two questions:   Is the above the correct way to get a vertical space in a table of contents? Why is it necessary to use the \protect command here?   Thank you... says:  SSH_ASKPASS    If  ssh needs a passphrase, it will read the passphrase from the    current terminal if it was run from a terminal.  If ssh does not    have  a  terminal associated with it but DISPLAY and SSH_ASKPASS    are set, it will execute the program  specified  by  SSH_ASKPASS    and open an X11 window to read the passphrase.   I'd like SSH to use an askpass program even if it was run from a terminal.  On occasion, I have to connect to servers, where there's some delay in showing a password prompt (maybe due to network issues, maybe due to attempted reverse DNS lookups, …). I get annoyed and switch to something else, and forget about the attempted connection. (Insert joke about attention span of a goldfish.) When I finally get back to it, the prompt's timed out and even a correct password would just result in a closed connection.  Keys would be one solution, but not every system I use has my usual SSH keys. However, I usually use Ubuntu systems, and Ubuntu has an SSH askpass program installed by default.  If an askpass window popped up, however, I'd be immediately aware of it. That's a good enough compromise for me, if only I can get it to work. Strange characters when the output of `ls` is redirected to a file I want to list the components of the current working directory in a text file  ls -1 &gt; textfile   Output looks fine with more.   1010661085645 1010729039145 1010747080245 1010849051345 1010859053445 1011046075845   However when I view this textfile with emacs, several strange characters appear  [0m[01;34m1010661085645[0m [01;34m1010729039145[0m [01;34m1010747080245[0m [01;34m1010849051345[0m [01;34m1010859053445[0m [01;34m1011046075845[0m   Can anyone explain what is going on here? How to sort a Google Sheets Query by a custom ordering Why is it bad practice to test for item names? I've heard that testing for item names and lores in Minecraft JE is bad practice. For example, the following commands are not good: execute if entity @e[nbt={Inventory:[{tag:{display:{Name:'[{&quot;text&quot;:&quot;Test Thing&quot;}]'}}}]}] clear @p minecraft:diamond{display:{Name:'{&quot;text&quot;:&quot;Shiny…&quot;}'}} 0 clear @p minecraft:netherite_sword{display:{Lore:['{&quot;text&quot;:&quot;A Mysterious Sword&quot;}']}} 0  Why are these commands considered bad practice? Are they prone to failure and may not work? And what can I do to fix them? Note that this only applies to reading item names, not setting them. The following are not considered bad practice and in fact are the only ways to accomplish these tasks: give @p minecraft:diamond{display:{Name:'{&quot;text&quot;:&quot;Shiny…&quot;}'}} replaceitem @p weapon.mainhand minecraft:netherite_sword{display:{Lore:['{&quot;text&quot;:&quot;A Mysterious Sword&quot;}']}} How to install gnome shell extensions offline? I know how to go to the  website and download gnome-shell extensions, but now I need to install some extensions available there on a computer without any internet access at all. It is in a internal corporate network and there's no way I can get outside internet access on it, so I must find another way. I can copy files in a usb disk.   At my home computer, I have found my extensions at ~/.local/share/gnome-shell/extensions/ but just copying this folder to the target corporate computer didn't do the trick.   Running gnome-tweak-tool gives me a "Install Shell Extension" button but I don't know how to download an extension in a format acceptable to install using this button. I have tried to point to the folder above but it didn't work either.  What do I need to do? Avoiding numerical cancellation question for $\sin x -\sin y$ for $x \approx y$ When trying to avoid cancellation, one tries to reformulate the equation in order to avoid subtraction between almost equal terms.   In $\sin (x) - \sin (y), x \approx y$ the suggested solution is to reformulate it to $$2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$  But I don't understand how it is any better? The subtraction between $x, y$ remains. Is it because $\lvert\,\sin(x)-\sin(y)\,\rvert\leq \lvert\,x-y\,\rvert$, so cancellation is less likely to happen between $x,y$ than the sines? Another (perhaps humorous) term for "The Gold Standard" Gold Standard as in:     an example against which others of the same type are compared   -    Any other humorous ways to refer to this?  Something like "the holy mother"... How to Upgrade Ubuntu 12.04.2, 12.04.3 to 12.04.4, the next Hardware Enablement Stack? I am currently using Ubuntu 12.04.2 32bit. I installed it using LiveCD.  Tomorrow, 23rd August 2013,Ubuntu 12.04.3 is released. I want to upgrade from Ubuntu 12.04.2 to Ubuntu 12.04.3 without using any LiveCD. Is it possible? If so please suggest me how can I do.  Actually while using Ubuntu 12.04.2 I have troubled with graphics. My graphics card is Intel i845G 64 MB. When Ubuntu starts it works fine and smooth without any lagging, but after sometime it hangs for few seconds (1 or 2 seconds) with a garbage screen and becomes sluggish. All windows and browsers start lagging and also it is not possible to play any video in any player (VLC, Movie Player, Xnoise, SMPlayer etc..).  I think Upgrading to Ubuntu 12.04.3 could fix my problem.  Any help will be greatly appreciated..    I've posted an answer to upgrade Ubuntu 12.04.x to 12.04.4. virtualbox crash on kernel 4.13.0-26 Ubuntu 16.04 LTS Virtualbox 5.0.40 ubuntu r 115130  Today after upgrading to kernel 4.13.0-26 guest vbox machines started to crash whole ubuntu. just hard poweroff is working.  If I startup with kernel 4.10.0-42 everything works fine.  what may be the problem ? Rectangles around cells in table Change Text Color in Blender Game Engine? I want to change the color of some text in my game. I've tried adding materials, but whenever I press P to play the game, the text stays the same white color. Any ideas? Linear algebra Upper Triangular matrix How to make dot match newline characters using regular expressions How do I install an older version of an app on an iPhone?