$\displaystyle\sum_{n=0}^{\infty}\frac{1}{(2n+1) . 4^n} $  Can anyone tell me how to approach? It must be convergent as $\displaystyle\sum_{n=0}^{\infty}\frac{1}{4^n}$ is convergent. and answer will be less than 4/3. but how to approach I can not see?
Evaluate $$ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$$  All i could do was to see that $$\frac{1}{3}=\frac{1}{2.1+1},\frac{1}{5}=\frac{1}{2.2+1},\frac{1}{7}=\frac{1}{2.3+1},\dots$$  SO, we should be able to write   $$ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots=1+\sum_{n=1}^{\infty}\frac{1}{2n+1}.\frac{1}{4^n}$$  This is what i could simplify the given question to...   But, this is doing me no good.. I could not go any further..  I would be thankful if some one can help me to clear this.  I would request users who are trying to help me to just give necessary hints but not post it as answer.  Thank You.
Suppose that a box contains 10 red balls, 20 green balls, and 30 blue balls.   Suppose also that balls are drawn from the box one at a time at random.  What is the probability that all the red balls are drawn before the blue or green balls are themselves exhausted. What is the probability that as the last red ball is drawn, there remains at least one blue and one green left in the box.  The answer I was given is $\dfrac{7}{12}$ and a general equation is: $$ \dfrac{b g}{1-b}+\dfrac{b g}{1-g} $$ where $$ g=\dfrac{20}{60},b=\dfrac{30}{60} $$ but why? I am having a lot of trouble with these series questions. Up until this point, I had relatively little trouble with all the questions in the book. These seem to require knowledge about approximations of functions and other external experience-based knowledge, which I just don't have yet.  Determine convergence or divergence of the given series. In the case of convergence, determine whether the series converges absolutely or conditionally.  $$\sum_{n=1}^\infty (-1)^n\left[e-\left(1+\frac 1 n \right)^n\right]$$  It's easy to see that  $$\lim_{n\to\infty}\left[e-\left(1+\frac 1 n\right)^n\right]=0$$  however, in order to apply Leibniz's Rule and show conditional convergence I need to show that the sequence is monotonically decreasing. This doesn't seem doable with straight inequalities, so I tried taking the derivative, which just resulted in an uninterpretable mess. This doesn't even begin to address the question of absolute convergence/divergence.  There are 54 of these questions... I must be missing something really fundamental if they all take this long. verbatiminput and fontsize How to unzip a zip file from the Terminal? Just downloaded a .zip file from the internet. I want to use the terminal to unzip the file. What is the correct way to do this? I read this book once when I was in middle school. A girl solves a murder or a mystery on a spaceship. It was a thin paperback books.   All I can remember is that there was an alien, and there was a movie star or actress or famous makeup artist on the ship traveling with another woman. By the end of the book the murderer was the woman the actress was traveling with who forced her to use her makeup skills to disguise her, and she was really some kind of alien. How to mount ufs file system under Debian testing? In order to mount my /dev/sda4  under /mnt/ufs_mount on Debian Testing , i have tried the following commands (it works fine under Debian Jessie):  modprobe ufs mount -r -t ufs -o ufstype=ufs2 /dev/sda4 /mnt/ufs_mount   and:  mount -t ufs -o ufstype=ufs2 /dev/sda4 /mnt/ufs_mount   But i get this error:  mount: /dev/sda4 is write-protected, mounting read-only mount: wrong fs type, bad option, bad superblock on /dev/sda4,    missing codepage or helper program, or other error     In some cases useful info is found in syslog - try    dmesg | tail or so.   The output of: dmesg | tail :  [ 1136.965142] ufs: ufs_fill_super(): bad magic number [ 1255.758946] ufs: ufs_fill_super(): bad magic number [ 2098.945757] ufs: ufs was compiled with read-only support, can't be mounted as read-write [ 2098.946045] ufs: You didn't specify the type of your ufs filesystem        mount -t ufs -o ufstype=sun|sunx86|44bsd|ufs2|5xbsd|old|hp|nextstep|nextstep-cd|openstep ...        &gt;&gt;&gt;WARNING&lt;&lt;&lt; Wrong ufstype may corrupt your filesystem, default is ufstype=old [ 2098.967212] ufs: ufs_fill_super(): bad magic number [ 2927.982112] perf: interrupt took too long (2504 &gt; 2500), lowering kernel.perf_event_max_sample_rate to 79750   Edit My sources.list:  deb http://httpredir.debian.org/debian/ stretch main deb-src http://httpredir.debian.org/debian/ stretch main  deb http://security.debian.org/debian-security stretch/updates main deb-src http://security.debian.org/debian-security stretch/updates main   And uname -a:  Linux debian 4.6.0-1-amd64 #1 SMP Debian 4.6.4-1 (2016-07-18) x86_64 GNU/Linux   How to mount ufs file system under Debian testing? I'tm trying to show in SQLcmd window the results of these queries:  SELECT SYSDATETIME() AS "date";  PRINT 'EXTRAÍDO POR:' SELECT SUSER_NAME() AS "Owner"  PRINT 'PASSWORD SETTINGS'; exec xp_loginconfig;  PRINT 'PASSWORD SETTINGS'; select * from sys.sql_logins;   And using this command in SQLcmd:  sqlcmd -S myServer -i inputpath -s, -o outputPath(like C:\log.xls) -e   So the sqlcmd output the file and does not show nothing in sqlcmd screen. How can i print these "Print" commands in the sqlcmd screen? (command promt).  When I run it on SSMS works well, like:  (1 row affected) OWNER:  (1 row affected) PASSWORD SETTINGS  (8 rows affected) PASSWORD SETTINGS  I would like to show in the prompt command this, like in the SSMS.  Thanks! Why is homology invariant under deformation retraction/homotopy equivalence? Is there some easy way to pad Strings in Java?  Seems like something that should be in some StringUtil-like API, but I can't find anything that does this. A club has $12$ members. They need to choose a programming team of $3$ members and a hacking team of $4$ members. Members can be on both teams, but Jack only wants to be on at most one team. How many ways can both teams be chosen?  I did two different ways. I solved if Jack chooses programming, $C(12,3)=220$ $C(11,4)=330$. In total there are $550$ different ways both teams can be picked.  But, if Jack chooses hacking, I solved $C(11,3)=165$ and $C(12,4)=495$. In total there are $650$ ways both teams can be chosen.  I was wondering if this is the correct method. I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. In this video, a college basketball player steals the ball, races down court, and avoids the defender by throwing a pass to himself off the backboard.    In all the (disorganized) basketball games I've played in, a shot had to hit the rim before the shooter was allowed to rebound. And that looks like a shot to me.  Obviously the officials allowed it, but I couldn't find an explicit mention in the . Is this ruled as a shot, and if so, why is it legal? I'd be interested in the answer for any level of basketball, though NCAA is the most relevant. Proving the measure of an increasing sequence of measurable sets is the limit of the measures I've made a simple snowfall particle simulation (using gandalf3's solution: ). Is there a way to loop the animation, so the transition between last frame and first frame is not visible? Is there a setting to do it? I installed the  on my Ubuntu&nbsp;11.04 (Natty Narwhal) with   sudo apt-get install lamp-server^   Now I navigate to /var/www and use   wget http://ftp.drupal.org/files/projects/drupal-7.x-dev.tar.gz   I get the following error.   --2011-08-03 13:59:00--   http://ftp.drupal.org/files/projects/drupal-7.x-dev.tar.gz Resolving  ftp.drupal.org... 64.50.233.100, 64.50.236.52 Connecting to  ftp.drupal.org|64.50.233.100|:80... connected. HTTP request sent,  awaiting response... 200 OK Length: 2757101 (2.6M)  [application/x-gzip] drupal-7.x-dev.tar.gz: Permission denied   Cannot write to `drupal-7.x-dev.tar.gz' (Permission denied).   How do I resolve this error?  I tried to download the same file in the downloads folder, and it went on without an issue. So it seems to be some permission error, but I'm not sure about that or how to solve it. nmake fatal error U1034: syntax error : separator missing How to differentiate formally? I have a Virtual machine using centos 7 with 200GB disk.For some reason my layout is:  df -h Filesystem               Size  Used Avail Use% Mounted on /dev/sda1                100GB  95GB  5GB      Running fdisk -l:  Disk /dev/sda: 204.8 GB, 204803670016 bytes,400007168 sectors  Units = sectors of 1 * 512 = 512 bytes  Sector size(logical/physical): 512 bytes / 512 bytes I/O size(minimum/optimal): 512 bytes / 512 bytes Disk label type: dos Disk identifier: 0x0006c283  Device Boot      Start         End      Blocks   Id  System /dev/sda1            2048   199997439    99997696     Linux   I would like to increase my space on my partision /dev/sda1. A 2D game called Deer Hunter has projectiles like this:    Are the projectiles 2D or 3D models? How could I implement something similar? Reference request: Standard textbook for first-order predicate logic in english I am writing my bachelor thesis in english, but I have studied logic in my different native language. Since a rely on elementary results from first-order predicate logic (especially satisfaction in structures), I should reference the reader to a textbook for details.   Is there a standard textbook that starts with elementary propositional logic and goes to Incompleteness theorems?