$ \sum{ a_n x^n } $ & $ \sum{ b_n x^n } $ converge at radiuses $ R_a, R_b $. What can be told about radius of $ \sum{(a_n+b_n) x^n} $? Problem: Suppose the power series $ \sum^\infty_{n=0}{ a_n x^n }  $ converges at radius of convergence $ R_a $ and the power series $ \sum^\infty_{n=0}{ b_n x^n }  $ converges at radius of convergence $ R_b $. What can be said about the radius of convergence of the power series $ \sum^\infty_{n=0}{ (a_n+b_n) x^n }  ~~ $ ? Hint: Handle the case $ R_a = R_b $ and the case $ R_a \neq R_b $ separately. Attempt- I didn't really know what to do but I still tried: If $ R_a \neq R_b $ the sum $ \sum^\infty_{n=0}{ (a_n+b_n) x^n } $ converges iff $ \sum^\infty_{n=0}{ a_n x^n }  $ and $ \sum^\infty_{n=0}{ b_n x^n }  $ converge, hence $ R_{ab} = min\{ R_a , R_b \} $. Let's now look at a case where we'd have $ R_a = R_b $; Define $ S^{a}_n = \sum^{n}_{k=1}(-1)^k x^k $ ,  $ S^{b}_n = \sum^{n}_{k=1}(-1)^{k+1} x^k $ note that $ R_a = R_b =1  $ . But $ S^{ab}_n = \sum^{n}_{k=1}( (-1)^k + (-1)^{k+1} ) x^k = 0 $ hence $ R_{ab} = +\infty $ . Question: I feel very lost at this problem, can you please help me as to what explicitly I should do and what to show? how would you approach this problem?
Radius of convergence of a sum of power series I have  two series  $\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$   $\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$  with radius of convergence $2$ and $3$ respectively.  How can I find the radius of convergence for the following?     $\displaystyle\sum_{n=1}^{\infty} (a_n +b_n)x^{n}$   Can anyone help me?
What is a good way to transform Cyclic Ordinal attributes? Your favourite application of the Baire Category Theorem How to prove useful property of logarithm of generating functional in QFT? How to prove that $\ln(Z(J))$ generates only? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point. Injecting beans into a class outside the Spring managed context I'm an end-user of one of my company's products. It is not very suitable for integration into Spring, however I am able to get a handle on the context and retrieve the required bean by name.  However, I would still like to know if it was possible to inject a bean into this class, even though the class is not managed by Spring itself.  Clarification: The same application which is managing the lifecycle of some class MyClass, is also managing the lifecycle of the Spring context. Spring does not have any knowledge of the instance of MyClass, and I would like to some how provide the instance to the context, but cannot create the instance in the context itself. Am I technically allowed to look up cards during a Magic tournament match? Am I allowed to look up a card during a game, if it's being played with serious Rules Enforcement? Either to read the full card from the Gatherer, or to try to find the exact card I'm vaguely thinking of.  Two possible scenarios:   An opponent has revealed a card previously, and I want to remember the exact details. (Something like the exact wording of a sorcery, mana cost, keywords on a creature, subtypes, etc. etc.) I have a "name a card and remove it from the opponent's deck" type card, and I want to, say, ask google for "Blue Gearhulk" to remember that it's that that one is called "Torrential Gearhulk".   Obviously, in casual or semi-casual games neither of these are a problem. Any reasonable opponent will just re-reveal the known card (assuming it IS still guaranteed to be known) or will accept "I name the Blue Gearhulk".  Also obviously, it seems unlikely that I (or anyone else) would get to the stage of playing at the REL without knowing the cards too thoroughly for this to be relevant.  But in principle... is it permitted? Or not? (and if not, is there a good reason, or just "we decided 'No'"?) I'm looking to have cell borders automatically applied when a cell is not empty. I thought I could use conditional formatting for this, but it doesn't seem to be an option. Numbering in algorithmicx I have simple code:  \documentclass{article}  \usepackage{algorithm}  \usepackage{algpseudocode}  \begin{document}  \begin{algorithm}   \caption{Examples}\label{alg:Examples}   \begin{algorithmic}[1]   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$    \end{algorithmic} \end{algorithm}          \end{document}     How to put a point after the number of the algorithm? I use chapters in report:  \documentclass[a4paper,12pt]{report}  \usepackage{algorithm}  \usepackage{algpseudocode}  \begin{document}  \chapter{Chapter 1}  \begin{equation} X_2=4 \end{equation}  \begin{algorithm}   \caption{Examples}\label{alg:Examples}   \begin{algorithmic}[1]   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$    \end{algorithmic} \end{algorithm}  \end{document}     How to change the numbering to take into account the chapters? Suppose I'm given the mean and one quantile (e.g. the 20% quantile) of a random variable $x$, and I want to find the parameters $\alpha$ and $\beta$ of a Beta distribution that has the same mean and quantile.  Is there an efficient way to do it?    Using just the mean, I know that since $\bar{x} = \frac{\alpha}{\alpha+\beta}$, we have $\beta = \frac{\alpha}{\bar{x}} - \alpha$.  So we only really have one parameter to estimate.  But I'm unsure how to use the quantile information to take the next step.  Maybe there's something I can do with the Incomplete Beta when I know the ratio $\frac{\beta}{\alpha} = \frac{1-\bar{x}}{\bar{x}}$?  I have access to R myself, so I could use a numerical optimizer for this, but ideally I need a method that can be carried out in Excel in someone else's environment.  Excel does have BETA.DIST() and BETA.INV() functions available.  A look-up table would be fine, but a closed-form formula would be better if it's possible. How do I know if an ability is magical? Often confusion arises when the  spell is used, an  is worn, or the 's anti-magic ray is employed, whether a given character's ability will still function.   The spell does give you some guidelines on how to adjudicate the effects of the loss of magic but says very little about how to determine what is affected.  In short, how do I tell if an ability is magical? Animations with movie15 and swf files I am trying to use movie15 to embed vector animations in my presentation. The trouble is that embedded SWF files refuse to display on Acrobat Reader X sometimes with a "This Media requires an additional player" message. Eg. this  doesn't work. Animated GIFs won't display either, with no error message.   Maybe this is an Acrobat issue but I tried the flashmovie package with the same SWF files and that does work (but only with the embedded player option so no controls are visible). But flashmovie is supposedly unstable with beamer so I was wondering if someone had run into the same problem with movie15.  I am using Windows 7, Acrobat 10.1.2 and MiKTeX. TikZ Angle Label - Help it "fit" better? I have the following drawn with TikZ    using the code  \documentclass{article} \usepackage{tikz} \usepackage{tkz-euclide} \usetkzobj{all} \begin{document} \begin{tikzpicture} \coordinate (O) at (0,0); \draw (O) circle (3); \coordinate[label = above left:$M$] (M) at (130:3); \coordinate[label = above:$N$] (N) at (80:3); \coordinate[label = below:$A$] (A) at (240:3); \coordinate[label =below right:$B$] (B) at (300:3); \draw (A) -- (M) -- (B); \draw (A) -- (N) -- (B); \tkzMarkAngle[draw = black, fill = white, opacity=1](A,M,B) \tkzLabelAngle[pos = 0.8](A,M,B){$x$} \tkzMarkAngle[pos = 0.2,draw = black, fill = white, opacity=1](A,N,B) \tkzLabelAngle [pos=0.8](A,N,B){$27 ^{\circ}$} \end{tikzpicture} \end{document}   Notice however, tht the 27 degrees does not fit in very nicely. How can I adapt the code to try and move the 27 a bit "lower" so that it fits in better, as well as the angle mark?  Also, how can I get two "double arrows" in the middle of lines AM and BN, to indicate that they are 'parallel'? First Order Time-Variant System : Picard Method I looking at a time variant first order system. I am trying to prove that a sequence of functions $x^{[k]}(t)$ generated using Picard iterations converges uniformly on some interval $[0,T]$.  Given:  $\dot{x}(t)=A(t)x(t)$ where $x(0)=x_0$  $x(t)=x∈R^n$ where $A(t)$ is a $n×n$ matrix  Assume there exists $M$ such that $||A(t)||\le M$ for all $t\ge 0$ and that $A(t)$ is a piecewise continuous function. Show that, for any $T &gt; 0$, a solution exists in the interval $[0,T]$.  I know that $x(t)=x_0+\int^t_0A(τ)x(τ) dτ$ must be satisfied if the solution does indeed exist.  I am supposed to use Picard iterations to prove this, which I am not familiar with yet. But I know that they are of the form: $x^{[k+1]}=x_0+\int^t_0A(τ)x^{[k]}(τ) dτ$. Where $x^{[0]}(t)=x_0$  Can I use the standard Picard proof to prove this or do I need to start somewhere else?  I was also asked to show that this method satisfies  $|x^{[k+1]}(t)-x^{[k]}(t)|\le |x_0| $$\frac{π^{k+1}(t)}{(k+1)!}$ where $π(t)=\int^t_0||A(τ)||dτ$.  I have no idea where to start with this, but hopefully it will become clearer once I finish the first part. What is the difference between Affection and Friendship, if any? I've risen Pokemon Amie Affection to 5 hearts on my Riolo in the hopes that it will evolve as soon as it levels up, but I'm not certain that this is the same thing as friendship, and I'd like to not waste a level.  My question is this: What is the difference between Pokemon Amie Affection and the Friendship stat? How can I search tags on StackOverflow Careers? I am attempting to search jobs by tag on StackOverflow Careers. On SO, this is done using square brackets, but that is simply doing a keyword search, and I still get results that don't have the tag I am looking for.  For example, when I search for the keyword "android", I get this:        which is the same as when I search for the keyword "[android]":        Is there a format I can use to search jobs by tag? If not, I believe this would be a nice feature (seems intuitive for the Stack Exchange invite-only job site). Optimization paramters read-only mysql 5.7 database? What's the best way to configure a MySQL database that is only read-only, and will never have any modifications made to it?  1) I have increased my innodb_buffer_pool_size to 70% of available RAM  2) Increase query cache size to 40mb  Running against a load generator, after some time the database starts bottlenecking to only 1000 statements per second.  The configuration is 2-cores, 8GB ram 50GB SSD. The database contains a single InnoDB 13 GB table with the same exact query running on two indexed columns.  We are using a vertx server to hit the database. The symptom we see is that the web service performance degrades after 15-20 minutes of heavy load.  Where can the bottleneck be? What parameters should we optimize? Is there a difference between DataTable.Clear and DataTable.Rows.Clear? T-SQL: Selecting Column Based on MAX(Other Column) Pullback of locally free sheaves is locally free states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module.     Suppose that $G$ is a locally free $O_Y$ module, hence it is a free $O_Y|_U=O_U$ module. We have an induced map  $$f^{-1}(U) \rightarrow U$$ given by restriction of morphism of ringed spaces.   Hence, as $\bigoplus O_U \simeq G|_U$, $f^*:Mod(O_U) \rightarrow Mod(O_{f^{-1}(U)})$ is functorial, $f^*(G|_U) = (f^*G)|_{f^{-1}(U)}$, we have isomoprhism,  $$ \bigoplus O_{f^{-1}(U)} \simeq (f^*G)|_{f^{-1}(U)} $$ Is there an operation similar to inset that creates faces in the opposite direction? That is, instead of creating faces inwards, they create faces outwards? A normal inset operation:  What I would like: How do I change the plymouth bootscreen?