Need a hint to compute $\displaystyle \sum_{x=0}^\infty xa^x$ and $\displaystyle \sum_{x=0}^\infty x^2a^x$, where $a \in (0,1)$.
I am very confused about how to compute  $$\sum_{n=0}^{\infty}nk^n.$$  Can anybody help me?
Please add TeX rendering in titles on the iOS app Subversion Obliterate feature I was just thinking of writing a shell script to implement the obliterate functionality in an easy to do way (externally, using the suggested way, but automated).  Here's what I had in mind:  On the client   svn list -R &gt; file-list. filter file-list in several ways like grep to create a file "files-to-delete", something like a set of grep XXX  file-list&gt;&gt;files-to-delete. transfer files-to-delete to the server using scp.   On the server   Dump the repository svnadmin dump /path/to/repos &gt; repos-dumpfile, this can be kept as a backup too. Filter the dump file, for each word in "files-to-delete", do: cat repos-dumpfile | svndumpfilter exclude $file &gt; new-dumpfile Create a new repository and load the new file to it svnadmin create new-name; svnadmin load new-name &lt; new-dumpfile   Would this work? How can it fail? Any other ideas? A probability theory question about independent coin tosses by two players Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice.    The way I thought of it is this : if Bob gets $0$ heads then there is no way he can get more heads than Alice. Otherwise the number of heads Bob can get which allows him to win is anything in the set $\{1,2,\dots,n+1\}$. And if Bob gets $x$ heads then the number of heads that Alice can get is anything in the set $\{0,1,2,..,x-1\}$. So\begin{align}P(\text{Bob gets more heads than Alice})&amp;= \sum_{x=1}^{n+1} \sum_{y=0}^{x-1} P( \text{Bob gets x heads }\cap \text{Alice gets y heads }) \\[0.2cm]&amp;= \sum_{x=1}^{n+1} \sum_{y=0}^{x-1} \left(C^{n+1}_x \frac{1}{2}^{x} \frac{1}{2}^{n+1-x}\right)\left( C^n_y \frac{1}{2}^y \frac {1}{2}^{n-y}\right)\\[0.2cm]&amp; = \sum_{x=1}^{n+1} \sum_{y=0}^{x-1} \frac{C^{n+1}_x C^n_y}{2^{2n+1}}\end{align}   How does one simplify this?      Apparently the answer is $\frac{1}{2}$ by an argument which looks like this, Since Bob tosses one more coin that Alice, it is impossible that they toss both the same number of heads and the same number of tails. So Bob tosses either more heads than Alice or more tails than Alice (but not both). Since the coins are fair, these events are equally likely by symmetry, so both events have probability 1/2. Selenium + FF : How to fix error, Unable to connect to host 127.0.0.1 on port 7055 after 45000 ms? My friend and his wife visited UK during the year 2015. And they came back on time we want to know one information as they same couple want to visit UK again  For how many years or months can multiple entry visa can be given for a couple who have already visited UK? Are "beginner questions" allowed on Stack Overflow? I noticed that  (about basic  syntax) received a lot of downvotes as soon as I posted it. Is there any specific policy against the posting of questions with "obvious" answers on Stack Overflow?  I'm unclear as to why questions with "obvious" answers aren't usually well-received by the community - are beginner questions considered harmful for some reason? ("Obvious" is a relative term, of course - this type of question would be helpful to anyone who was learning C# for the first time, but it wouldn't be helpful to an experienced C# developer.) Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$ I'm stuck in full screen in the node editor. Escape isn't working, and I have no menus and no idea how to leave except to quit Blender and reopen the program. Relative performance of std::vector vs. std::list vs. std::slist? Prove $\lim\limits_{n\to\infty}n^{-1/(p+1)}(1^{1^p}\cdot 2^{2^p}\cdots n^{n^p})^{n^{-p-1}}=e^{-1/(p+1)^2}$ Prove that for real number $p\geq  0$ we have   $$\lim_{n\to\infty}\frac{(1^{1^p} \cdot2^{2^p}\cdots n^{n^p})^\frac{1}{n^{p+1}}}{n^\frac{1}{(p+1)}} = \exp\left(\dfrac{-1}{(p+1)^2}\right).$$ no border around hyperlinks with hyperref Given a set $\mathcal{S}$ of size $n$, let a $k$-partition of $\mathcal{S}$ be a grouping into $k$ disjoint classes $$(S_1 ,S_2,...,S_k)$$   Where the $S_i$ do not necessarily contain the same number of elements.  Let $C(S_1, S_2, ... S_k)$ be the set of combinations which can be made by taking exactly one element of each class, (i.e. $C(S_1, S_2, ... S_k)$ is the Cartesian Product $S_1 \times S_2 \times ...\times S_k$). For a given $(n,k)$, what is the minimum number of k-partitions $N$ such that  $\bigcup\limits_{i=1}^{N} C(S_1^{(i)}, S_2^{(i)}, ...,S_3^{(i)}) = $ All Possible Combinations  Example:  for $n=4$ and $k=2$, $\mathcal{S} = \{1,2,3,4\}$ we can choose:  $$ S^{(1)} = \left\{\left\{ 1, 2\right\}, \left\{ 3, 4\right\}\right\}$$ $$ S^{(2)} = \left\{\left\{ 1, 3\right\}, \left\{ 2, 4\right\}\right\}$$  Then $$C_{1} = \left\{\left\{ 1, 3\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 2, 4\right\}\right\}$$ $$C_{2} = \left\{\left\{ 1, 2\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 3, 4\right\}\right\}$$  and $C_1 \cup C_2 =$ All possible combinations and $N=2$. The startup disk cannot be partitioned or restored to a single partition. Windows 10 and Mojave The startup disk cannot be partitioned or restored to a single   partition.      The startup disk must be formatted as a single Mac OS Extended   (Journaled) volume or already partitioned by Boot Camp Assistant for   installing Windows.   I had installed Windows 10 via bootcamp in my mac. Current version is 10.14.1  Now I don't want windows and need that space back but getting above error when I go bootcamp to remove it.  This is the output of diskutil list  /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         181.3 GB   disk0s2    3:       Microsoft Basic Data BOOTCAMP                60.3 GB    disk0s3    4:           Windows Recovery                         509.6 MB   disk0s4  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +181.3 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      4.3 GB     disk1s4   And this is of diskutil cs list  No CoreStorage logical volume groups found   Can anybody please help me to solve this problem?  Output after I tried commands given in the first answer  ❯ diskutil list /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         250.8 GB   disk0s2  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +250.8 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      3.2 GB     disk1s4 How to get relative path from absolute path There's a part in my apps that displays the file path loaded by the user through OpenFileDialog. It's taking up too much space to display the whole path, but I don't want to display only the filename as it might be ambiguous. So I would prefer to show the file path relative to the assembly/exe directory.  For example, the assembly resides at C:\Program Files\Dummy Folder\MyProgram and the file at C:\Program Files\Dummy Folder\MyProgram\Data\datafile1.dat then I would like it to show .\Data\datafile1.dat. If the file is in C:\Program Files\Dummy Folder\datafile1.dat, then I would want ..\datafile1.dat. But if the file is at the root directory or 1 directory below root, then display the full path.   What solution would you recommend? Regex?  Basically I want to display useful file path info without taking too much screen space.  EDIT: Just to clarify a little bit more. The purpose of this solution is to help user or myself knowing which file did I loaded last and roughly from which directory was it from. I'm using a readonly textbox to display the path. Most of the time, the file path is much longer than the display space of the textbox. The path is supposed to be informative but not important enough as to take up more screen space.  Alex Brault comment was good, so is Jonathan Leffler. The Win32 function provided by DavidK only help with part of the problem, not the whole of it, but thanks anyway. As for James Newton-King solution, I'll give it a try later when I'm free. How to delete OS from boot menu After installing fedora and Centos and other OS now I have them all showing in my Boot menu [F2] on start up not the menu form the OS even after deleting them and reinstaling other OS - Ubuntu -so It just shows one when the OS starts but when I go to UEFI setting I still find all the old ones they instruct you to use DEL to take them out but that does not work Show community wikis differently in the Android app Badass and BAMF are both modern words with approximately the same meaning: "Someone who is awesome to an extreme level, thereby leveraging unquestionable authority." Is there another noun or title that would convey the same level of awe, admiration and toughness that calling someone a badass does, without the vulgarity?  Thanks for any help. :) Is it possible to trade CFD without leverage? I'm new to trading. Based on what I read, trading CFD involves high risk because of leverage.   Is it possible to trade CFD stocks without leverage (in Etoro for example)?  If this is possible, then I would lose/gain only the percentage by which the stock goes down/up.  Thank you "Neither Michael nor Albert is correct" or "Neither Michael nor Albert are correct"? What is the correct sentence?  Neither Michael nor Albert is correct.   Neither Michael nor Albert are correct. Unfortunately MyApp has stopped. How can I solve this?