I'm trying to calculate the following series: $$ \sum_{n=1}^\infty \frac{1}{3^n-2^n}=\frac{1}{3-2}+\frac{1}{9-4}+\ldots $$ or the general case: $$ \sum_{n=1}^\infty \frac{1}{(x+1)^n-x^n} $$ I have tried some methods but still have no idea, is there any way to solve such series?
I have to give low and high bounds for the following:  $$ \sum_{n=1}^\infty \frac{1}{2^n - 3^n } $$  How do I determine an upper bound? How can I show this sum exists?  edit: removed erroneous conclusion.
Reprojected MODIS NDVI has range from -32768 to 32767, expected -1 to 1 This is the problem I am currently having trouble with:  If $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix  $$         \begin{bmatrix}         a &amp; b\\         c &amp; d\\         \end{bmatrix}$$prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$.     $$A=\begin{bmatrix}     a^2 &amp; ab &amp; b^2\\     2ac &amp; ad+bc &amp;2bd\\     c^2 &amp; cd &amp;d^2\\     \end{bmatrix}$$  What have I considered so far:   I tried to find characteristic polynomial of first matrix and then find the roots which would be $\lambda_1$ and $\lambda_2$. I was hoping I can make a connection with characteristic polynomial of matrix $A$, but it got too complicated and did not seem like the right idea. Since there are two eigenvalues for first matrix, there are two eigenvectors. I thought about somehow using $A\vec v_i=\lambda_i\vec v_i$, where $i=1,2$  Also, I thought about finding a connection between these two matrices. What I see is that the sum of elements of first and second column, of first matrix, are $(a+c)$ and $(b+d)$, respectively.  Sum of first, second and third column of matrix $A$ are $(a+c)^2, (a+c)(b+d)$ and $(b+d)^2$. Now, if I could assign $\lambda_1$ and $\lambda_2$ to $(a+c)$ and $(b+d)$, then I would get that sum of first, second and third column of matrix $A$ are $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$, but what would be next?   I thought it would be good to combine 2. and 3rd idea, but I am not making any progress.  I appreciate any help. How long does the Truffle take to move in after building the mushroom biome? When should "into" be used rather than "in to," and vice versa? "Into" (one word) and "in to" (two words) are frequently confused.  In what situations should the former be used?  The latter? The  claims that likelihood and probability are distinct concepts.     In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a clear distinction in perspective: the number that is the probability of some observed outcomes given a set of parameter values is regarded as the likelihood of the set of parameter values given the observed outcomes.    Can someone give a more down-to-earth description of what this means?  In addition, some examples of how "probability" and "likelihood" disagree would be nice. Difference between "asked" and "were to ask" in conditional sentences? Synchronize Clocks I need to develop a system where there are 4 sensor boxes powered by USB and they have their internal clock. Each of these boxes are connected to a Linux pc and all these PCs are linked with WiFi. API of the sensor box allows me to read their internal precision clock (uSec accuracy).   I like to synchronize all four clocks of the sensor boxes as close as possible. Hopefully within a few micro seconds of each other and compensate for drift over time. The idea is to use the sensor data that is collected at the same time (The data I receive is time stamped with the boxes' internal clock). The latency on usb bus, WiFi, pc timing are all indeterministic.  Is this possible given my setup? pgfplots: Clipping Everything Outside a Specific Area I want to add information to each column of a histogram. every node near coord is probably the best approach. It works well with bar plots but not with the histogram:  \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{compat=newest} \usetikzlibrary{pgfplots.statistics}  \begin{document}  \begin{tikzpicture} \begin{axis}[ ybar interval, xticklabel=\pgfmathprintnumber\tick--\pgfmathprintnumber\nexttick, % Code to generate labels nodes near coords={ \pgfmathprintnumber{\pgfkeysvalueof{/data point/y}} }, every node near coord/.style={ anchor=south,xshift=9mm, }, % ] \addplot+[hist={bins=3}] table[row sep=\\,y index=0] { data\\ 1\\ 2\\ 1\\ 5\\ 4\\ 10\\ 7\\ 10\\ 9\\ 8\\ 9\\ 9\\ }; \end{axis} \end{tikzpicture}  \end{document}   I used every node near coord/.style to shift the labels.  The problem is that I have four labels for three columns/bars:    I addition I would like to center the labels. A workaround will be a manual adjustment (every node near coord/.style) just like in the shown example.  Update  Maybe clipping is a possibility. Is there a way to clip the labels (in this case the 7 in the right) away?  Like this: Passing data between view controllers There are several questions here about , about , and about . And of course there's the basic . And plenty of  questions. There's related questions like . But there's no be-all-end-all question asking:  What exactly determines depth of field in a photograph?   Is it just a property of the lens? Can lenses be designed to give more depth of field for the same aperture and focal length? Does it change with camera sensor size? Does it change with print size? How do those last two relate? Hide textfield blinking cursor What happens if you call erase() on a map element while iterating from begin to end? In the following code I loop through a map and test if an element needs to be erased.  Is it safe to erase the element and keep iterating or do I need to collect the keys in another container and do a second loop to call the erase()?  map&lt;string, SerialdMsg::SerialFunction_t&gt;::iterator pm_it; for (pm_it = port_map.begin(); pm_it != port_map.end(); pm_it++) {     if (pm_it-&gt;second == delete_this_id) {         port_map.erase(pm_it-&gt;first);     } }   UPDATE: Of course, I then  which I didn't think would be related but answers my question. What formula should be used to determine "hot" questions? Why use a z test rather than a t test with proportional data? I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question.  Prove directly from the definition of Ito's integral that   $$\int_0^t B_s^2 dB_s=\frac{1}{3}B_t^3-\int_0^t B_s ds$$  I have tried for a couple of hours but I simply cannot prove it . I would be grateful if somebody could help me out , even a hint would be very helpful Isomorphism between two models of Hilbert's Axioms with different notions of congruence I've been reading about Hilbert's Axioms of geometry, and I have two models on $\mathbb{R}^2$ which satisfy the usual axioms of incidence, betweenness, and congruence.   The first model has a distance function given by  $$ d(A,B)=|a_1-b_1|+|a_2-b_2| $$ where $A=(a_1,a_2)$, $B=(b_1,b_2)$, and two segments $AB\cong CD$ if $d(A,B)=d(C,D)$.  The second model has a distance function given by $$ d(A,B)=\sup\{|a_1-b_1|,|a_2-b_2|\}, $$ and again $AB\cong CD$ if $d(A,B)=d(C,D)$.  The author claims that these two models are isomorphic in that there is a bijection between the sets of points in $\mathbb{R}^2$ that preserves lines, betweenness and congruence. I've been toying with it for a while, but it seems to me that the notions of congruence in particular are too different.   For example, in the first model, if $A=(0,0)$ and $B=(2,1)$, and $C=(2,0)$ and $D=(5/2,5/2)$, then $AB\cong CD$ since both segments have length $3$. To preserve congruence, I suppose two segments are congruent if the maximum of the displacement along the $x$-axis or $y$-axis by traveling along one segment is equal to the maximum of the displacement along either axis when traveling along the other segment. In this sense, how could you shape the segment $CD$ so that the maximum of the displacements along either axis is the same as the maximum of the displacement for whatever the image of $AB$ is?  In the first model, $AB$ is congruent to any segment which is the hypotenuse of triangle with legs that sum to $3$, and this sum could split into any two terms. I just don't see a function that could handle all these possibilities at once. Does anyone know of a particular isomorphism between these two? Thanks. When does a Möbius transformation map $\Im(z)>0$ to itself? Show Möbius transformation which maps $\Im(z)&gt;0$ to itself iff  $$ f(z)= \frac{az+b}{cz+d}\,,\,\,ad-bc&gt;0$$ and $a,b,c,d$ are real. What could cause java.lang.reflect.InvocationTargetException? How do I flatten this vector logo leaving only the visible black paths? Basic desktop actions are not available on Ubuntu 20.04 In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window.