$$\sum_{n=1}^{N}\frac {z^n}{2^n}$$   I attempted to use the method of difference but it didn't work.
Why does the following hold:  \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*}  Can we generalize the above to      $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?   Are there some values of $x$ for which the above formula is invalid?  What about if we take only a finite number of terms? Is there a simpler formula?     $\displaystyle \sum_{n=0}^{N} x^n$   Is there a name for such a sequence?    This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: .
Captured variable in a loop in C# novel with psi slavery Batch file to create Folder based on Date and Copy Files from One Location to Another Location? How can I convert monsters from D&D 3 or 3.5 to 2ed monsters? Why are floating point numbers inaccurate? Subsetting matrices Setting up different application instances under one DNS Linux - Help, I'm running out of inodes! I have a MySQL table with utf8 general ci collation. In the table, I can see two entries:  abad abád  I am using a query that looks like this:  SELECT *  FROM `words` WHERE `word` = 'abád'   The query result gives both words:  abad abád  Is there a way to indicate that I only want MySQL to find the accented word? I want the query to only return    abád    I have also tried this query:  SELECT *  FROM `words` WHERE BINARY `word` = 'abád'   It gives me no results. Thank you for the help. When you are viewing an app in Google Play store, it displays the following : Updated, Size, Installs, Current Version, Requires Android, and Rating.  I am trying to do some research and figure out the initial release / publish dates of different apps. Is there a way to get this information from Google Play, or perhaps a website with this information? I have 5 tables and a section after them. After creation of an output file I have two tables on one page, then a section and last 3 tables on next page.  What I want to achieve is to put tables and section below them.  \begin{table}[ht]\footnotesize \centering \begin{tabular}{|ll|} ...... \end{tabular} \label{tab:result} \end{table}  \section{Section below tables} How to add a modifier to a collection? My goal is to reduce the faces of an object using the decimate modifier. The object I'm editing has lots of vertex groups. Blender 2.8 only lets me add the decimate modifier to a single group at once which would take quite some time. I have already put all groups in a single collection.  How can I add the decimate modifier to the entire collection?  Do I have to add the modifier group by group? How to Avoid Precision Errors While Creating Numpy Arrays? When I double-click on a script in Nautilus to run it, the script just opens in my text editor with no option to run it. Using Nautilus, how do I run executable text files and/or scripts? Accepting an answer should automatically upvote it as well Equivalent of *Nix 'which' command in PowerShell? How do I ask PowerShell where something is?    For instance, "which notepad" and it returns the directory where the notepad.exe is run from according to the current paths. how to make some border lines of a table thick and colored Is there an easy way to calculate the total cost of tolls for a trip within the United States? Is there a convenient way to calculate the cost of tolls between two arbitrary points within the United States? I'm planning to take a trip along the Eastern seaboard and will be passing through multiple states with different toll-collecting authorities. The mapping and trip planning resources I've found so far will only indicate whether a particular section of road has tolls, not the price of those tolls. I'm aware that the various toll-collecting authorities post their rates online, but these charts are often difficult to use if you're not already familiar with the area. Isn't there an easier way? VirtualBox: using physical partition as virtual drive Background: I am using VirtualBox installed on Windows 7. From within VirtualBox I am using Xubuntu as a virtual OS. The reason I chose this approach is so that I don't have to keep turning off Windows and rebooting from Xubuntu every time I needed to switch OSes. And VirtualBox's seamless mode is pretty amazing to allow me see Xubuntu and Windows 7 all in one screen.   Issue: Now I am thinking of a way to have Xubuntu more integrated into my system. By this I mean I want to have a physical partition for Xubuntu. But I want to still have the feeling of the seamless mode.  Question: So finally, my question is: is it possible to load a partition in VirtualBox as a virtual OS?   Case examples: Ideal scenario would be: I physically boot up and login to Windows 7. Now I want to access Xubuntu, so I load VirtualBox and access my Xubuntu partition without rebooting. And the other way around too, i.e. I boot up the system, login to Xubuntu, and can access the actual Windows 7 partition through VirtualBox.   Other info: Please note that I am not talking about getting access to files, as I have a completely separate partition for my files, and am very familiar with VirtualBox's Shared Folders option. How do I install the latest stable version of Thunderbird?