Suppose $(x_n)$ is a convergent sequence such that $x_n&gt;0 \forall n \in \mathbb{N}$. Set $$y_n=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}, n\in\mathbb{N}$$ Prove $\lim_{n\to\infty}{x_n}=\lim_{n\to\infty}{y_n}$ &lt;- How should I approach this? Also prove that if $x_n$ diverges to $+\infty$, $(y_n)$ diverges to $+\infty$ &lt;- I suppose I should be able to do this if I prove the above statement?
Let ($x_s$) be a convergent sequence, where $x_s&gt;0$ for all s, and $y_s$ be a sequence such that   $$\displaystyle \Large y_s=\frac{s}{\frac{1}{x_1}+...+\frac{1}{x_s}}$$  Prove $\displaystyle \lim x_s$=$\lim y_s$  So I know things like $x_s$ is bounded, and that it's Cauchy, but I can't think of anything to solve it.  Any help is appreciated.
What does the number next to the review link in the top bar actually mean? May I use Google Fonts for creating a logo with commercial purposes? I want to create a logo for a contest, the winner will be paid and the winning logo will be the institutional logo of an academic entity, so I need a font which can be used freely or at a cheap cost for commercial purposes.  I see that Google Fonts are delivered as "free fonts" for websites:  May I use them freely for graphic design products (a logo in particular) as well? How to create a node with submissions of a webform? I'm using Drupal 7 and created a content type 'NewArticle' with the fields 'Title', 'Text' and 'Image'. Users have the permission to create a 'NewArticle', but I don't like the way of displaying the administration bar in order to let them create a 'NewArticle'.  Instead, I'd like to work with . My goal is to create a Webform with exactly the same fields as the content type 'NewArticle'. So what I want is: when users filled in the form and submit it, a new node 'NewArticle' is created with the submitted information.  How can I do this? Are there modules that make this possible? I can't find any... Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When? where is `cd` located? Recursively rename files and directories I already have the "Edit Questions And Answers" privilege. Yet, if I approve someone else's edit, I'm told that more votes will be required before the edit is accepted.  This seems a bit peculiar to me, since I could just copy the edited version and paste it myself. Language Options -> Spell Checking in Windows 10 version 1803 Does standard c library provides linked list etc. data structures? While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? Just a curiosity. 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How many sequences of $n$ tosses of a coin that do not contain two consecutive heads have tails as the first toss? If you toss a coin $n$ times, there are $2^n$ possible sequences of heads and tails. Let $E_n$ be the set of sequences which do NOT contain two consecutive heads and $e_n$, the number of sequences in $E_n$.  Thus $E_3$ $=$ $\{$ $TTT$, $TTH$, $THT$, $HTT$, $HTH$ $\}$ and $e_3$ $=$ $3$.  How many elements of $E_n$ have $T$ as the first toss?  Please can anyone help me out here? Undefined reference to static class member Can anyone explain why following code won't compile? At least on g++ 4.2.4.  And more interesting, why it will compile when I cast MEMBER to int?  #include &lt;vector&gt;  class Foo {   public:       static const int MEMBER = 1;   };  int main(){       vector&lt;int&gt; v;       v.push_back( Foo::MEMBER );       // undefined reference to `Foo::MEMBER'     v.push_back( (int) Foo::MEMBER ); // OK       return 0; } What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information.