Evaluating $\sum_{n=1}^{\infty}\frac{(n-1)!}{(x+1)(x+2)(x+3)\cdots(x+n)}$ for positive real $x$ Let $x$ be a positive real number. Evaluate the infinite sum $$\sum_{n=1}^{\infty}\frac{(n-1)!}{(x+1)(x+2)(x+3)\cdots(x+n)}$$ My approach was writing the expression as $$ x!\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)\cdots(n+x)}$$ Then I assumed $x=2$ then I got answer as $1/2$, so the answer was $1/x$ (for general). But can I get any idea for a generalized solution of the question?
Evaluating $\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$ for $x\in\mathbb{R}^{+}$ I am trying to evaluate the following sum. Here $x\in\mathbb{R^{+}}$. $$f(x)=\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$$ At present I do not know how to proceed. Any hints are appreciated. Thanks.
Are proofs by contradiction really logical? Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.   My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?  That, to me, seems the exact opposite of logical. It sounds like we say "so, I'll have a really big problem if this thing isn't true, so out of convenience, I am just going to act like it's true". When I try to bring down the Achievements tab in Area 51, it's showing an error dialogue.  Strangely enough, the global inbox dropdown works just fine.  Could be connected to .  Screenshot: Can't install apps from App Store on iPhone with iOS 7 Collection of short stories; in one a spy attempts to escape detection on Mars's moon by moving faster than the spaceship can scan Probability that a quadratic polynomial with random coefficients has real roots The following is a homework question for which I am asking guidance.     Let $A$, $B$, $C$ be independent random variables uniformly distributed between $(0,1)$. What is the probability that the polynomial $Ax^2 + Bx + C$ has real roots?   That means I need $P(B^2 -4AC \geq 0$). I've tried calling $X=B^2 -4AC$ and finding $1-F_X(0)$, where $F$ is the cumulative distribution function.  I have two problems with this approach. First, I'm having trouble determining the product of two uniform random variables. We haven't been taught anything like this in class, and couldn't find anything like it on Sheldon Ross' Introduction to Probability Models.  Second, this strategy just seems wrong, because it involves so many steps and subjects we haven't seen in class. Even if I calculate the product of $A$ and $C$, I'll still have to square $B$, multiply $AC$ by four and then subtract those results. It's too much for a homework question. I'm hoping there might be an easier way. Short story about a group of sci-fi writers sitting around discussing their profession I have 14.04.5/16.04 or newer and an AMD GPU. I'm not getting the performance I want, and some programs say they need things like OpenCL.  I've heard about fglrx, but if I try to install it on my machine, it breaks and I have no GUI.  Is there anything I can do to get my AMD card working to its full potential? I was wondering why I couldn't delete my own question. Well actually, I have accepted the question itself to be wrong and even I feel that I have accepted a wrong answer, but when I try to delete my question, I get a message saying that,"This question can't be deleted, flag it for moderator attention". Why is this happening and how can I solve this? Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$ Solve $\sum nx^n$ I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert&lt;1$.  This solution makes sense to me:  $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n = \frac{d}{d x} (1-x)^{-1} \\ \sum_{n\ge0} nx^{n-1} = (1-x)^{-2} \\ x \sum_{n\ge0} n x^{n-1} = x(1-x)^{-2} \\ \sum_{n\ge0} nx^n=\frac x{(1-x)^2}$  However, a  I am reading used the following method:  $$\sum_{n\ge0}nx^n=\sum_{n\ge0}x\frac d{dx}x^n= x\frac d{dx}\sum\limits_{n\ge0}x^n=x\frac d{dx}\frac1{1-x}=\frac x{(1-x)^2}$$  This seems closely related to the solution I described above, but I am having difficulty understanding it.  Can someone explain the method being used here? How to add the name of the supervisor in a @thesis field? In the biblatex doc, it says ( Version 1.7, p.11):     thesis      A thesis written for an educational institution to satisfy the   requirements for a degree. Use the type field to specify the type of   thesis.      Required fields: author, title, type, institution, year/date      Optional fields: subtitle, titleaddon, language, note, location,   month, isbn, chapter, pages, pagetotal, addendum, pubstate, doi,   eprint, eprintclass, eprinttype, url, urldate   I would like to know if there is any "canonical" way to add the name of the supervisor(s) in a field of the .bib file (no .bbl manipulation please, I want my bibliography to be usable "as it" in several documents).  The expect result could be something like  :     Meand Myself. “A brilliant result, explained”. PhD thesis**, under the   joint supervision of Pr. Toto and Pr. Tata**. Université Paris 24,   Nov. 2013.   The "under the (joint) supervision" could be a string that would be declined in four ways:   under the supervision of  under the joint supervision of and the abridged forms I remember reading a story about an alien who came to earth and gave/sold a human a little device.  It was the size of a pen, and when you aimed it at something metal, the metal became malleable, like clay.  A second time, and the metal resolidified.  When the alien came back not long after, human civilization had completely fallen to pieces because of it.  Does anyone know the name and author of the story? Why does "under the sea" mean "in the sea"? When I input [asdf][lisp] into the search field on , without being logged in, only one tag remains. Is it broken? Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point? Is the function $f(x) = \lim_{k \to +\infty} \tan{kx}$ a right example?  But I do not know whether this is a function at first. Is this a function?  Could you give a correct real-valued function that is continuous at precisely one point?  Thanks. Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$ How can we find the exact value of the infinite sum  $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$  This problem appears in:  , p.114. How can I add a program to the context menu of all files? Alien invasion movie, giant robot aliens with long legs capturing people So I saw this movie about 10 years ago, when I was about 5 years old. Since I was really young, my memory of it isn't really that great.   I think it may have been an alien invasion movie, but the only scene I remember is that there were these giant robot-alien things:   I think they had several long legs. They had a big glass dome on top of them They would grab civilians and drop them in their glass dome.   That's all I remember. If you could please help me out, that would be great.   Again, I was about 5 years old when I watched it, so my memory of it isn't that great, but I hope you know what I'm talking about. Thanks so much :) Ignore python multiple return value Say I have a Python function that returns multiple values in a tuple:  def func():     return 1, 2   Is there a nice way to ignore one of the results rather than just assigning to a temporary variable? Say if I was only interested in the first value, is there a better way than this:  x, temp = func() What does 'per unit frequency' refer to? We were discussing Kirchoff's theorem on blackbody radiation in class today and the equation was: $$e_{f}= J(f,T)$$  $e_f$ was defined as power per unit area per unit frequency. What does per unit frequency refer to here? I know what per unit time means but don't understand 'per unit frequency'.