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?
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.
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This is its link,    I already try Alt button + ASCII code number of this type of special characters, from numpad key, just example rename file/folder and press Alt+60 (for &lt; characters) Still it giving message that, A filename cannot contain any of the following characters: \ / : * ? " &lt; > |   Suggest me suitable solution so that I can give name of my file/folder with this type of special characters. we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically   but the question is, what is mathematics ? how can we define it as a branch of science ? Prevent suggested edits under 10 reputation on protected questions Recently, . The specific  is protected:  protected by  ♦ Apr 27 '11 at 1:10 This question is protected to prevent &quot;thanks!&quot;, &quot;me too!&quot;, or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.  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Considering this code, can I be absolutely sure that the finally block always executes, no matter what something() is?  try {       something();       return success;   }   catch (Exception e) {        return failure;   }   finally {       System.out.println("I don't know if this will get printed out"); } Select 1px aliased border When I choose Select > Modify > Border and select a 1px border - the border I get is anti-aliased, hence if I fill my selection a 3px border is affected.  Is there a way to select a 1px aliased border? When is the limit of a sum equal to the sum of limits? I was trying to solve a problem and got stuck at the following step:  Suppose ${n \to \infty}$ .  $$\lim \limits_{n \to \infty} \frac{n^3}{n^3} = 1$$  Let us rewrite $n^3=n \cdot n^2$ as $n^2 + n^2 + n^2 + n^2 \dots +n^2$,$\space$ n times.  Now we have  $$\lim \limits_{n \to \infty} \frac{n^3}{n^3} = \frac {n^2 + n^2 + n^2 + n^2 + n^2 \dots +n^2}{n^3} $$  As far as I understand, we can always rewrite the limit of a sum as the sum of limits ...  $$\dots = \lim \limits_{n \to \infty} \left(\frac{n^2}{n^3} + \frac{n^2}{n^3} + \dots + \frac{n^2}{n^3}\right)$$  ...but we can only let ${n \to \infty}$ and calculate the limit if all of the individual limits are of defined form (is this correct?). That would be the case here, so we have:  $= \dots \lim \limits_{n \to \infty} \left(\frac{1}{n} + \frac{1}{n} + \dots + \frac{1}{n}\right) =$[ letting ${n \to \infty}]$ $= 0 + 0 + \dots + 0 = 0$  and the results we get are not the same.  Where did I go wrong?