I have to show that $\sum_{i=1}^{\infty} \frac{A}{(1+r)^i} = \frac{A}{r}$. Where $ -1 &lt; r &lt; 1$. I tried putting it like: $$\sum_{i=1}^{\infty} \frac{A}{(1+r)^i} = \frac{A}{r+1} + \sum_{i=2}^{\infty} \frac{A}{(1+r)^i}$$ However I don't know how to continue from here on.  This is not a duplicate, most other questions require $|1+r|&gt;1$ which is not necessarily the case here. I would be very grateful if you could help me out or give me pointers how to solve this!
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: .
I want to do something like:  MyObject myObj = GetMyObj(); // Create and fill a new object MyObject newObj = myObj.Clone();   And then make changes to the new object that are not reflected in the original object.  I don't often need this functionality, so when it's been necessary, I've resorted to creating a new object and then copying each property individually, but it always leaves me with the feeling that there is a better or more elegant way of handling the situation.  How can I clone or deep copy an object so that the cloned object can be modified without any changes being reflected in the original object? Do I need a visa to visit the UK with my EU partner? I am not from EU. I have Spanish partner and we live in Spain and, because of our official relationship, I have a Spanish family resident permit. We are going to travel to the UK together for tourism. I am confused whether I need an EEA family permit or not.  I will not live there and I'm going to travel with my partner. I was told that I don't need a visa, that showing our civil union document will be sufficent. Is this correct, or do I need a visa to enter the UK?    Thank you so much for your answer. Let me give you more detail.  I hold Turkish citizenship.  My resident card exactly says that Residence Card of a Family Member of a Union Citizen " tarjeta de familiar de la union.   my case is like For example, a non-EEA spouse of a German national living in Germany will usually hold a residence permit issued under German domestic law. Therefore, a United Kingdom EEA family permit is required for travel and entry to the UK , therefore I should consider that I need to EEA Family Permit even if for touristic trip with my partner. In Drupal 7, I used the Secure login module to run the site on a secure connection, but it seems it doesn't revert back to http on logout. I would like to just run the whole site as https:// as a simplification.  Is there a simple way to do this without the use of module? How do I set the default program? How do I set the default program that I use?  I have tried to open System Info → Default Program → change my program, but it won't work,  Any idea? Or should I use Terminal?   I want to replace the default movie player with VLC media player, because the current movie player is useless to me. Syntax of bash alias definition I'm using Cygwin, and have installed all the developer packages as well as most if not all of the other packages. When I provide an alias for gcc, I get the error  -bash: alias: gcc: not found   What could I be doing wrong?  I am typing the following command into the terminal:  alias gcc "gcc -ansi -Wall -g -O0 -Wwrite-strings -Wshadow -pedantic-errors -fstack-protector-all"   FYI, this alias works on other systems. Outlier Detection on skewed Distributions Under a classical definition of an outlier as a data point outide the 1.5* IQR from the upper or lower quartile, there is an assumption of a non-skewed distribution. For skewed distributions (Exponential, Poisson, Geometric, etc) is the best way to detect an outlier by analyzing a transform of the original function?  For example, distributions loosely governed by an exponential distribution, could be transformed with a log function - at which point is it acceptable to look for outliers based on the same IQR definition? How does the British-Irish Visa Scheme (BIVS) work? states:  "In practice, explicit right outer joins are rarely used, since they can always be replaced with left outer joins and provide no additional functionality."  Can anyone provide a situation where they have preferred to use the RIGHT notation, and why? I can't think of a reason to ever use it. To me, it wouldn't ever make things more clear.  Edit: I'm an Oracle veteran making the New Year's Resolution to wean myself from the (+) syntax. I want to do it right What is the correspondence between the conserved canonical energy-momentum tensor, which is $$ T^{\mu\nu}_{can} := \sum_{i=1}^N\frac{\delta\mathcal{L}_{Matter}}{\delta(\partial_\mu f_i)}\partial^\nu f_i - \eta^{\mu\nu}\mathcal{L}$$ (the four conserved Noether currents corresponding to four possible spacetime translations)  where $\{f_i\}_{i=1}^N$ are the $N$ matter fields in the theory, and we assume $f_i\mapsto\alpha^\nu\partial_\nu f_i$ for translations,  and stress-energy tensor from the Einstein-Hilbert action, which is: $$ T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{Matter}}{\delta g^{\mu\nu}} $$   In particular, how do you get that the two are equal (are they?) for Minkowski space, for which there is no variation in the metric? Issues with win key and right alt key after Windows Update (Polish diacritical signs not working) I understand that multivariate regression models contain more than 1 dependent variable, but what is the difference between running a multivariate regression with dependent a and b to a set of independent variables, rather than running two regression models, one for a and one for b against the same set of independent variables? Why is the area under a curve the integral? I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.  However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the anti-derivative. Can someone explain why that is the definition of the integral and how Newton figured this out? As a follow up to my other , I would like to know whether the following is possible:  I have $20,000 in my brokerage account The price of a call option with $300 strike price is $5.00 (1 contract = 100 shares) I buy 20 contracts of these $300 calls (cost is 20 * 5 * 100 = $10,000) The price of the underlying goes up, making my option in-the-money  I decide to exercise my options. Exercising my calls means that I buy 20 * 100 = 2,000 shares of the underlying at $300 per share. However, I only have $10,000 left in my brokerage account and am unable to pay for the shares (2,000 * 300 = $600,000). What now? How does this work? Do brokers recognize that my position is profitable, and buy and sell the underlying for me as soon as I exercise my calls? In other words, do I need to worry about being able to pay for the shares of the underlying if I decide to exercise my options? Can I just take my profit without bothering with the shares at all? How to type directly with keyboard the U+200C character? How to type directly with keyboard the U+200C character in Notepad or other places... in Windows 8.1? (without depending on a certain font)  I don't want to copy and paste ways. I don't want any complicated ways. I just want something like Alt+### for typing U+200C Directly at any text area in windows 8.1 without depending to a certain font. Just like ALT+129 for U+200B.  (I do not need ALT+129, That's for U+200B. I need U+200C)  the U+200C character name is: ZERO WIDTH NON-JOINER (ZWNJ)  Here is the U200C character info: When do relativistic effects need to be explicitly included? hyperref: how to format text in a forms text-field I would like to pre-format the text in a form text-field, e.g. make it bold, italic or use a specific font. The following code generates a simple text-field:  \documentclass{scrreprt} \usepackage{hyperref} \begin{document} \textbf{The Text Field} \begin{Form} \TextField[]{\ } \end{Form} \end{document}   Is it possible to pre-format the text in the text-field? The hyperref package manual (10/2011) speaks of an optional parameter format and the use of JavaScript to format the field. Can this be used to format the text in the field? And if yes, how is it done? (I am not proficient in JavaScript) GNU screen: Running commands in different window when screen is started Ways to evaluate $\int \sec \theta \, \mathrm d \theta$ Making a live USB with persistent storage What are the badges I can earn on each site, and what are the exact criteria for earning each badge?