How should I rewrite the sequence so that we have a form that we can easily calculate the limit?
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
Eigenvalues of symmetric matrices are real without (!) complex numbers Is there any proof of the fact that the eigenvalues of symmetric matrices (i.e. $A\in\mathbb{R}^{n\times n}$ with $A^t=A$) are real without the use of the concept of complex numbers? Where are Upstart log messages on Ubuntu 13.X? Why is my profile messed up on a single site and how can I revert it? I was just asking a question on Gardening SE and suddenly my user name there is a25bedc5-3d09-41b8-82fb-ea6c353d75aand my picture is replaced by a Gravatar. My location is also left blank.  Here's a   Curiously, my profiles on all other sites are fine.  I don't have a separate login for SE, I log in using my Google account. Nobody other than me has access to my Google account. I use two step authentication and I can see on "recent devices" in Google that no one but me has accessed my Google account. I only access SE from home, and nobody other than me has used my devices.   I tried to edit my Gardening SE profile back to the way I want it to be and I got the following message:     Oops! There was a problem updating your profile:   Display name may only be changed once every 30 days; you may change again on Jun 20 at 10:49   I'm assuming this is because I changed the "about me" on all SE sites two days ago.  I never edit my profiles individually. I always apply changes to all sites, using the same profile on all sites.  I have certainly not made such strange (or any) edits to my profile on Gardening SE.  Why did this happen, and how can I get my proper user name and picture back on Gardening SE? Do I really have to wait for a whole month? I want to use the site, and I want them to recognize me as the user they know. I don't want to wait a whole month.  Update: The Gardening SE moderator Niall C. helped me out, and he offered the following potentially relevant information:     I changed your display name on Gardening &amp; Landscaping back to "Fiksdal". Your profile history shows that your profile image (from your headshot on imgur.com to a gravatar), email, about me also changed at the same time as your display name on May 21. It also shows that I changed your profile image to a different gravatar when I changed your display name just now... weird! – Niall C. 1 hour ago I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a series of algebraic manipulations that can lead from $\sum\limits_{i=0}^n \binom{n}{i}$ to $2^n$. What is the difference between "INNER JOIN" and "OUTER JOIN"? Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.  For the P $\implies$ Q part, I know you can get a surjection $Y\to X$ by mapping $y$ to $x$ if $y=f(x)$ for some $x\in X$ and mapping $y$ to some arbitrary $\alpha\in X$ if $y\in Y\setminus f(X)$. But I don't know about the Q $\implies$ P part.  Could someone give an elementary proof of the theorem? Shapefile layer no longer editable (toggle edit option greyed out) in QGIS? computing the orbits for a group action Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then the number of orbits for the action of $G$ on the fields with nine elements is   3 5 6 9   I have no idea how to compute the numbers of orbits for a group action. Anyone please help me. Thanks. Convergence of $\sum _{n=1}^{\infty }\left\{ 1-n\log \frac {2n+1} {2n-1}\right\} $ My created page with link  The 404 page after following my new url. PC automatically booting into windows 10 in dual boot I have both windows 10 and ubuntu 16 installed on my system.   Lately the system is automatically booting to windows. To boot to ubuntu, I first have to pause startup by pressing escape and then boot into ubuntu.  I know that I have to change the boot order to Ubuntu  but am not able to do so and the grub customiser helps to change only the boot order in grub menu to the best of my knowledge.  What should I do? I am unable to find any way to change the boot order in bios Solve $\sin(z) = z$ in complex numbers Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot! From the  article on Wikipedia, the following with regard to the Coriolis Effect on a rotating sphere:     By setting vn = 0, it can be seen immediately that (for positive φ and ω) a movement due east results in an acceleration due south. Similarly, setting ve = 0, it is seen that a movement due north results in an acceleration due east. In general, observed horizontally, looking along the direction of the movement causing the acceleration, the acceleration always is turned 90° to the right and of the same size regardless of the horizontal orientation.   My intuitive (but possibly incorrect) understanding is that if there are two points, Point A and Point B, at different latitudes in the Northern hemisphere, the Eastward velocities of these points are different because they are at different distances from the Earth's axis of rotation, and this causes the Coriolis effect for a rotating sphere.  If a projectile is fired due North from Point A near the Equator towards Point B near the North Pole, the projectile will start off with the higher Eastward velocity of Point A, and will land to the East of Point B, which is moving East at a slower velocity than the projectile.  Firstly, is that correct?  If that IS correct, that brings me to the quote off Wikipedia. The quote implies that, if a projectile is fired due East, it will experience a Coriolis force to the South. My intuitive explanation based on differences in velocities between origin and destination does not account for a Southwards movement at all, since the Point A and Point B velocities are identical if the points are on the same latitude.  What am I missing? Would a projectile fired due East or due West experience any North or South drift caused by the Coriolis effect (or anything else for that matter)?  Why? I keep hearing a lot about functors in C++. Can someone give me an overview as to what they are and in what cases they would be useful? I'm referring to the colored circles with numbers next to them on the top of the page, to the right of my reputation counter. What do they represent? Solving an Op-amp circuit The image of the circuit is shown below and it required to find \$V_0\$,  My first attempt at solving this problem is by changing the current source into a voltage source with 1-V and 2k\$\Omega\$ resistance. The fact that the inverting and non-inverting terminals aren't grounded make this problem look difficult.To the point which, I don't know how to proceed with this question or where to start. I would appreciate any help. Following some thought and another schematic from a hint suggested by Alfred, I produced a schematic representing our work.  And  my solution for the problem is below, Using node equation at nodes A and B we have, $$\frac{V_A-V_B}{1k}=-0.5 \text{mA}$$ $$\frac{V_B-(2+V_A)}{1k}=-0.5-x$$ where x is the current that is sent in the output of the op-amp. Using KCL, at the bottom node near the current source we see that the same current that goes through the op-amp also goes through the \$2\text{k}\Omega\$ resistor. Hence, we have, $$\frac{-V_A}{2k}=x$$ Replacing this in the second equation, $$\frac{V_B-(2+V_A)}{1k}=-0.5+\frac{V_A}{2k}$$ And and solving the equations yields \$V_A=-2 \text{-V}\$ and \$V_B=-1.5 \text{-V}\$ Why did my electric range wiring burn out at the terminal block? Turned on our 2 year old stove (Whirlpool Gold) and heard fizzing noises, then smelled electrical burning. Went downstairs and shut off the breaker (40 amp). That was the only way to get it to stop "fizzing". (6 gauge wire by the way). The range is 240 volt electric 3 wire hooked up to a 40 amp breaker (6 gauge wire). When I took the back panel off the stove I saw exactly where the problem occurred. The black wire was burned, the terminal block was melted, and the inside of the back metal panel that I removed was pitch black in the spot that was facing that area before removal. I removed the stove top and inspected all wires and other electronic components. No problems at all.   So my questions are:    Why did the black wire only, at the terminator block, start burning?  Why did the breaker not pop off?  Is it safe to just replace the terminator block or is this an indicator of a much larger problem?  Is 6 gauge the correct gauge for a 40 amp breaker? Can I dump my non-Retina Images for a iOS7 only app? Is there any way to accept only numeric values in a JTextField? Rounding Percentages