I need to find all values of $a,b&gt;0$ for which the series converge. I know that a geometric series converges if $|r| &lt; 1$ but how can I find all the values for which the series converges in general?  $$ \sum_{n=0}^{\infty}(a^{n} - b^{n}) $$
We let a>0 and b>0 and  we have $\sum_{n=0}^{\infty}b^n-a^n$. We have to find all values a,b>0 where $\sum_{n=0}^{\infty}b^n-a^n$ is convergent. I think I can use that $$\sum_{n=0}^{\infty} b_n &lt; \infty , \sum_{n=0}^{\infty} a_n &lt; \infty \implies \sum_{n=0}^{\infty} (b_n-a_n) &lt; \infty. $$. And I think $\sum_{n=0}^{\infty}b^n$ and $\sum_{n=0}^{\infty}a^n$ is convergent if a&lt;1 and b&lt;1? It will make sense. But how can I show it?
Drush PHP syntax error when running I am following the instructions to install Drush that are listed on this page:   I seem to be getting a syntax error within the code:  [user@localhost ~]$ wget https://github.com/drush-ops/drush/releases/download/8.0.0-rc4/drush.phar --2015-11-16 16:23:02--  https://github.com/drush-ops/drush/releases/download/8.0.0-rc4/drush.phar Resolving github.com (github.com)... 192.30.252.131 Connecting to github.com (github.com)|192.30.252.131|:443... connected. HTTP request sent, awaiting response... 302 Found 2015-11-16 16:23:05 (1.47 MB/s) - drush.phar saved [3496991/3496991]  [user@localhost ~]$ php drush.phar core-status  Parse error: syntax error, unexpected '[' in phar:///home/user/drush.phar/includes/output.inc on line 188   I am not seeing this error on Google. Any idea how to fix it? How can I get rid of those little white dots? I am using Blender 2.70a. I have been using Ubuntu 12.04 for quite a while and have never had a problem come at me out of the blue like this. I'm trying to install python 3.2 on my Ubuntu system and every time I run the code to compile the python source, I get this error:  'Command 'sudo' is available in '/usr/bin/sudo' The command could not be located because '/usr/bin' is not included in the PATH environment variable. sudo: command not found'   I've also tried running gksudo command to view what my environment shows but I pretty much get the same error. I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this method works, and I just can't seem to see anything, although I know what type of intuition I am trying to make out.   So, my question is why we can use the division algorithm to convert between the decimal bases and any other number bases?   An example from the decimal system to binary would be appreciated. Asymptotic expansion of $u_{n + 1} = \frac12 \arctan(u_n)$ Since I am a regular Stack Overflow follower, I came across this and was shocked a bit regarding the design and user experience perspective.  Attaching picture 1 - it states that I can click any site and view a synopsis about it. Clear!!!    Attaching Picture 2 - you can see that when I clicked Webmasters, the design wants me to scroll horizontally to view all information and basically it hides information.    It would be nice, if it did something like below : What's wrong with "I'll open you the door"? When I call the buzzer outside my girlfriend's flat, she sometimes says *"I'll open you the door". I correct this to "I'll open the door for you".  I've never heard a native speaker say it the first way, which is why I think it's wrong. But I can't explain why.  There is a pattern in lots of English phrases that would suggest both are correct. A few common examples:   Give the keys to me before you go — Give me the keys before you go I'll buy a coffee for you at the cafe — I'll buy you a coffee at the cafe We sent a text to Martin on his birthday — We sent Martin a text on his birthday   The left-hand side is more formal, and the right-hand side is more common in everyday speech.  What do you call this pattern?  What makes the 'open-the-door' sentence an exception? How to combine bold and "code sample" in Markdown? How can I make a word in a code sample bold? I only see asterisks around the expected bold text.  sample **sample** sample Is using passive voice "bad form"? Give an example of a sequence of real numbers with subsequences converging to every real number Prove: The pre-image of an ideal is an ideal. Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$. What is the status of existing measurements of the speed of gravity? Is Angular momentum conserved in Impure rolling No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$ There is a proof in my textbook for the following claim, which doesn't make a whole lot of sense to me.  My annotations are in bold.  Could someone perhaps elaborate on what's going on?  Claim.  If there does not exist a cycle containing edges $e$ and $g$ then there exists a vertex $u \in V (G)$ such that every path in $G$ sharing one end with e and another with $g$ contains $u$.  Proof: The claim trivially holds if $e$ or $g$ is a loop OK, so we assume that neither is. Let $P$ with vertex set $v_1, v_2, . . . , v_k,$ in order, be a path with $e$ joining $v_1$ to $v_2$ and $g$ joining $v_{k−1}$ and $v_k$. Let $f_i \in E(P_i)$ be the edge with ends $v_i$ and $v_{i+1}$. Let $j$ be chosen minimum so that no cycle in $G$ contains $e$ and $f_j$ We can do this because we know that at least the edge $g$ will not create a cycle by assumption, right?. We will show that $u = v_j$ satisfies the claim.  Suppose not. Let $C$ be a cycle containing $e$ and $f_{j−1}$ What if $f_j = e$?  then what cycle? a single vertex? and let $P′$ be a path from an end of $e$ to an end of $f$ Not sure what the book meant by $f$ here, any guesses? avoiding $u$. Choose a subpath $Q$ of $P′$ with one end in $V (C)$ and another in ${v_{j+1}, v_{j+2}, . . . , v_k}$ as short as possible. Then $C \cup Q \cup P$ contains a cycle containing both $e$ and $f_j$, a contradiction. (The last statement requires some case checking.) This ending seems a bit abrupt and non-obvious to me  Thanks for the help How does Austria's suspension of Schengen affect travellers (both EU and non-EU)? I'm an Italian citizen. Next Sunday I'm going to Austria. I have a national ID card but no passport. Recently Austria suspended Schengen. So can I enter Austria? What does the notation like 8.6e-28 mean? What is the 'e' for? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist Question:  a. Find an example for a non-negative and continuous function s.t. $\int _0^\infty f(x)dx$ is finite but the following limit doesn't exist: $\lim_{x\to \infty} f(x)$.  b. Is it possible that $\int _0^\infty f(x)dx$ is bounded but $f(x)$ is not bounded?  What we did with A: We suggested the function $|sin(x^2)|$ which has periods that get smaller and smaller until they no longer imply on the sum. But since it's a trig function it doesn't have a determinate limit. Wolfram didn't have the integral value for that func, and we were wondering if it really converges and if it is really a good example.  What we did with B: We thought about the function: $f(x)=  { x\in \Bbb N },{e^{-x} \notin \Bbb N }$ and we had a disagreement whether f(x) is integrable. I said no because similarly to Dirichlet function, one sum's limit will be 0 while the other one's will be infinite. My partner said that if I find a $\delta&lt;1$ then the definition of Riemann's integral does hold and so this integral is equal to that of $f(x)=e^{-x}$ She was trying to say this functions integral will be 0, but still it won't be bounded. I disagreed. Who's right? Increment Record based on field I have Solaris 8 OS installed in French.  Is it possible to switch from French to English without reinstalling the OS ? If yes how ?  When locale -a i have this :  POSIX common en_US.UTF-8 C iso_8859_1 en_CA en_CA.ISO8859-1 en_US en_US.ISO8859-1 en_US.ISO8859-15 en_US.ISO8859-15@euro fr_CA fr_CA.ISO8859-1 iso_8859_15 th th_TH th_TH.TIS620   Even when I try to edit /etc/default/init with :  LANG=en_US.ISO8859-15 LC_COLLATE=en_US.ISO8859-15 LC_CTYPE=en_US.ISO8859-15 LC_MESSAGES=en_US.ISO8859-15 LC_MONETARY=en_US.ISO8859-15 LC_NUMERIC=en_US.ISO8859-15 LC_TIME=en_US.ISO8859-15 LC_ALL=en_US.ISO8859-15   and reboot : nothing happens.  To verify the result i always use date and when i reboot : i also have the date in french.  P.S : I have to do this configuration in a computer which isn't here in my office, so i have to this manipulation in a an another computer here in my office and if it works, i'll do it to the other computer.  So the computer here is configured in English and i have to switch it to French. Here is the content of my ~./dt :  C                     errorlog              sessionlogs Desktop               errorlog.old          sessions Trash                 errorlog.older        simulateur.dtwmrc appconfig             help                  startlog appmanager            icons                 startlog.old asr.dtwmrc            outils.dtwmrc         startlog.older bin                   palettes              tmp bookmarks             sdtfonts              types dtwmrc                sdtpcaction.conf      wsmenu en_US.ISO8859-15      sdtpcaction.conf.bak   Thanks How can I avoid "please insert CD/DVD" error when updating Ubuntu? I can't update 12.04 LTS. The Update Manager shows this:  CD/DVD 'Ubuntu 12.04 LTS _Precise Pangolin_ - Release i386 (20120423)' is required Please insert the above CD/DVD into the drive '/media/cdrom/' to install software packages from it.   The image I downloaded and burned to CD is from  the PC (Intel x86) desktop CD version. I burned many CDs but the result is the same.