I'm trying to find the sum of $$\sum_{n=0}^∞ \frac{(3n)}{4^n}(-1)^n$$ So far I've tried finding the nth partial sum and finding the sum through the geometric series equation k/(1-r) to no avail.
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
The number of different ways to choose $k$ out of $n$ unique items The number of different ways to choose $k$ out of $n$ unique items:   Without repetition and without order-significance: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ Without repetition and with order-significance: $\binom{n}{k}\times k!=\frac{n!}{(n-k)!}$ With repetition and with order-significance: $n^k$ With repetition and without order-significance: $?$   Thank you! Based on the question , I got curious. Is there a guide or resource that describes how the price of a ticket is determined?  When you search you often see curious effects like cheaper tickets for a flight from A to B with a layover compared to a direct flight. This seems counter-intuitive from an outsider's perspective since you fly twice. Also one way tickets more expensive than 2 ways.  It can be a short set of rules. Something that helps someone to get started with finding good prices. I know these can be very complex and they probably change over time and over airline interests but there is probably a simplified set of rules that can be enunciated. mod_security: another rule with same ID I have installed httpd 2.2.15 on CentOS 6.5 (minimal installation with no cPanel) with modsecurity 2.8. When I am starting httpd I am getting this error:  Starting httpd: Syntax error on line 23  of /etc/httpd/conf.d/modsecurity.conf:  ModSecurity: Found another rule with the same id   The line 23 is:  "id:'200000',phase:1,t:none,t:lowercase,pass,nolog,ctl:requestBodyProcessor=XML" From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.  I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation.  See also: Google search results showing up a URL on its search which doesn't exist on the website I am using a keyword to display a website page on google search. After entering the keyword, on the first page of google, I am redirected to a page, whose domain is exactly similar to the website whose page I want to display, but it is showing a page which doesn't even exist on the domain. Also, as similar domain with spam characteristics is also getting displayed.   To practically demonstrate this issue, do follow the below steps.   Go to  Type dolce streichquartett and click on search. The 4th/5th URL has this link http://www.dolce-streichquartett.de/?high_ry=/casinos/de/spielautomaten-casino-bregenz.php, which is no where on the website and looks like spam The 7th/8th link you see is http://deonlinecasino.org/doc/kostenlos-roulette-spielen-spielgeld-ausdrucken.php, which is exactly similar to the above with the title as the keyword but a different domain.   What is this issue and why is this happening? The MySQL manual at  covers this.  Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently RENAME {DATABASE | SCHEMA} db_name TO new_db_name; .  This needs to work with , which stores things very differently than . How can I convert my Java program to an .exe file? Baking cloth simulation use only 20% of cpu ressources, how to get 100%? Baking a cloth use only 20% of CPU, inactive process of cpu is 78%, is there a way to use those extra 78% ? Blender 2.75 Windows 10 Can't view deleted questions via Android app If I have a 18-55mm lens, is there a point in buying a 35mm prime lens? What is a smart pointer and when should I use one? What is a smart pointer and when should I use one? What is the VTable Layout and VTable Pointer Location in C++ Objects in GCC 3.x and 4.x? Setting proxy for apt from terminal Regression: Transforming Variables How to compress / reduce the size of JPEG photos for archiving? I have 50,000 high resolution JPEG photos, where a couple of them might occasionally be needed, about once a year.  I wanted to zip them to save disk space, except that zipping gives no space benefit - so trying to reduce the images disk usage using winzip, winrar or 7zip was not successful.  Is there any software or algorithm similar to zip to compress image size on hard disk for storage without loosing any image information? Fine, the 's math  did in the old pop-up. That makes sense. But this is just silly. Why do I need to be notified that a post I downvoted got deleted?    (The title links next to the red -1s were removed from the screenshot by me. They're present in the real pop-up.)  So I can break out a bottle of the '82 Rothschild? It's especially pointless because there's  that caused the change.  Please don't display the Rep Signal when the change is only due to someone else's deleted posts. Creaking from cranks/spindle. How to fix? What is the best possible way to check if a string can be represented as a number in Python?   The function I currently have right now is:  def is_number(s):     try:         float(s)         return True     except ValueError:         return False   Which, not only is ugly and slow, seems clunky. However I haven't found a better method because calling float in the main function is even worse. How to manually typeset text in parallel on facing pages in (Lua)LaTeX? I want to typeset a bilingual text in parallel on facing pages using LuaLaTeX. The existing solutions to automate this task have proven unsatisfactory as they don't allow for footnotes (parallel), don't allow for different font sizes of the two parallel texts (), or generally provide a poor result of aligned text chunks (eledpar).  As the text is not too long, I really came to consider breaking the pages myself. For this to do, I have to solve the following problem(s):  Consider two facing pages with ending paragraphs of different length. The following paragraphs should then begin at the same vertical position such that the beginnings of these following paragraphs face each other. For this to accomplish on has   to determine the position of the end of the longer paragraph on its page to have the means to let the next parallel paragraphs both begin at the identified position.   Is this something which is doable? Are there mechanisms provided by (Lua)LaTeX to determine positions on a page and to begin typesetting at a specific position on a page?  EDIT: The following example illustrates the task. Here the latin paragraph starting with "O tempora, o mores" (third chunk of text in the code) should begin at the facing position of the beginning of the paragraph "Shame on the age..." (fifth chunk).  \documentclass[a6paper]{scrbook} \usepackage{fontspec} \usepackage[latin,english]{babel} \begin{document} \mbox{}\pagebreak  \noindent\selectlanguage{latin}{\large quo usque tandem abutere, Catilina, patientia nostra? quam diu etiam furor iste tuus nos eludet? quem ad finem sese effrenata iactabit audacia? nihilne te nocturnum praesidium Palati, nihil urbis vigiliae, nihil timor populi, nihil concursus bonorum omnium, nihil hic munitissimus habendi senatus locus, nihil horum ora voltusque moverunt? patere tua consilia non sentis, constrictam iam horum omnium scientia teneri coniurationem tuam non vides? quid proxima, quid superiore nocte egeris, ubi fueris, quos convocaveris, quid consili ceperis quem nostrum ignorare arbitraris?\par\newpage}  \noindent\selectlanguage{english}When, O Catiline, do you mean to cease abusing our patience? How long is that madness of yours still to mock us? When is there to be an end of that unbridled audacity of yours, swaggering about as it does now?  Do not the nightly guards placed on the Palatine Hill — do not the watches posted throughout the city — does not the alarm of the people, and the union of all good men — does not the precaution taken of assembling the senate in this most defensible place — do not the looks and countenances of this venerable body here present, have any effect upon you? Do you not feel that your plans are detected? Do you not see that your conspiracy is already arrested and rendered powerless by the knowledge which every one here possesses of it? What is there that you did last night, what the night before — where is it that you were — who was there that you summoned to meet you — what design was there which was adopted by you, with{\parfillskip0pt\relax\par}  \noindent\selectlanguage{latin}{\large O tempora, o mores!\par\newpage}  \noindent\selectlanguage{english}which you think that any one of us is unacquainted?  Shame on the age and on its principles! \end{document} The Follow Questions and Answers feature is now live across the Network