My attempt 1: Let $x_n=\left(1+\frac{1}{2^2} + \cdots + \frac{1}{n^2} \right)$, and we have $x_{n+1}&gt;x_n$. Since $$ 1+\frac{1}{n^2} \le 1+ \frac{1}{(n-1)(n+1)} = \frac{n}{n-1} \cdot \frac{n}{n+1}, $$ then $ x_n &lt; \frac{2n}{n+1} &lt;2$. Hence $\{x_n\}_{n=2}^{\infty}$ converges. Let $$ \lim_{n \to \infty} x_n =a, $$ and notice that $x_{n+1}=\left(1+\frac{1}{(n+1)^2}\right)x_n $. By this, we can only get $a=a$. We can't know the value of $a$.  My attempt 2: We write $$ \prod_{k=2}^n \left(1+\frac{1}{k^2} \right)=\exp\left(\sum_{k=2}^n \log\left(1+\frac{1}{k^2}\right)\right) $$ So it suffices to know what's the limit of the serise on the right. I still don't know how to finish it.  Finally,I used Mathematica to calculate the limit, and it tells me that is $\frac{\sinh(\pi)}{2\pi}$. But I don't know how to know it without computer. Can you help me?
How would I go about finding $\prod_{n=1}^\infty (1+\frac{1}{n^2})$ ? Setting $a_k = \prod_{n=1}^k (1+\frac{1}{n^2})$, the first few partial products are $a_1 = 2$, $a_1 = 2.5$, $a_3 \approx 2.78$, $a_4 \approx 2.591$, $a_{20} \approx 3.50$, and they eventually appear to converge at around $3.639$. (These values were found with a quick python program. So much for my initial conjecture that the product converged to $e$.)   I tried writing $1+\frac{1}{n^2}$ as $\frac{n^2 + 1}{n^2}$, but it wasn't immediately obvious how to prove convergence / find the limit because the product doesn't really factor or telescope. Is there a way to calculate the limit?
example of not surjective isometry on a bounded not closed set. Is there an example of a bounded not closed set $A\subset \mathbb R^n$ for some $n\in \mathbb N$ and an isometry $f:A\rightarrow A$ ($\forall x,y\in A\space;\space d(x,y)=d(f(x),f(y))$ where $d$ is the euclidean metric) such that $f$ isn't surjective? calculating expected number of packets. Change equation font in Pages During a Pages update which was pushed out a few months ago, Pages obtained the ability to write equations using LaTeX or MathML commands. According to an Apple Support article, the size and color of the equation can be changed ():  The equation appears at the insertion point in your document (or before the selected text). The equation is an inline object, the same size and color as the surrounding text. If you change the size or color of the surrounding text, the size and color of the equation’s font also change.  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Please comment on this. I would really like to see all of my questions and answers on the profile page, even if some of them were deleted, and I don't have enough rep to see them on the site. (Note that some questions are  after 30 days or 1 year, and the author might be oblivious about what happened.)   , the &quot;deleted recent [questions/answers]&quot; links in one's own user profile now list all respective posts that were deleted in the past 60 days, rather than just those that were originally posted in the last 60 days. , users with the &quot;access to moderator tools&quot; privilege (10k+ on designed sites, 2k+ on beta and non-designed sites) can use the search operator deleted:1 to see all their own deleted posts. , deleted questions and answers that were posted in the past 60 days, can be seen using the &quot;deleted recent questions&quot; and &quot;deleted recent answers&quot; links on the questions and answers tabs (at the bottom of the tab) in your user profile. For older posts: as of 2013-04-23 you can  if you already have a link to them, but they still aren't linked from the user profiles (not even just for you). Nor do you see inbox notifications for comments on them if you come back a few hours later. You have to have thought to bookmark your question, or you have to go digging for it in your browser history. Where are the NetworkManager logs? A previous poster on the regular forums (  ) said /var/log/daemon.log, but I don't seem to have such a file. Do I need to enable logging? When to delete your 0 vote answers? I've been in the SO community a little over a month now - and having a great time!  As I grow into a contributing citizen of stackoverflow, i'd like to ensure that i'm being responsible with my answers.   I've answered a few questions, some without upvotes, some voted on and some accepted.  I feel that after a certain period of time, 0 vote answers are probably providing more clutter than value.  Apparently, nobody has found them useful, so they probably aren't (or are poorly written). Tonight I went through and deleted my answers with 0 upvotes where some answer was accepted.  That was an easy decision.   I left my 0 vote answers where nothing was accepted yet, well, because I think they could provide the information that the OP requested.  But at some point, I think my answers might just be cluttering the issue.  So my question to you is: How do you know when it's time to delete your 0 vote answers?  Or even 1 vote answers for that matter? How to prove this for positive real numbers? $$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$  I tried AM-GM, CS inequality but all failed. Find duplicate files Ubuntu 12.04 gnome issue, no clock is shown, strange starter icon cannot be removed i am finding nowhere a solution I am running UBUNTU 12.04 and using GNOME desktop, which I installed additionally. Logging into classic GNOME I see the following:     WHAT is this "earth globe" icon coming from?Seems to be named "starter". It is configured to open a website, from my bookmarks. WHY is it not showing the TIME?   Using dconf-editor I had no luck in answering the above questions.  Any idea? I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ  \cdots  \circ {\tau _n}(z)$  By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n {} ,\bigcup\limits_{i = 1}^n {} ,\bigcap\limits_{i = 1}^n {} ,$ I want to write ${T_n}(z) = \left( {\mathop  \circ \limits_{i = 0}^n {\tau _i}} \right)(z)$ or even ${T_n}(z) = {\mathop  \circ \limits_{i = 0}^n {\tau _i}} (z)$. Can I do this?