I'm reading about infinite products in complex analysis, where there is a theorem like     The product $\prod_{n=1}^\infty\left(1+a_n\right)$ converges absolutely iff the series $\sum_{n=1}^\infty|a_n|$ converges.   Then an exercise is to show that $\prod_{n=2}^\infty\left(1-\dfrac{1}{n^2}\right)=\dfrac{1}{2}$  The theorem above guarantees that the product converges, but what is the method to evaluate its value?
Evaluate the infinite product $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$  I can't see anything in this limit , so help me please.
Find all continuous functions satisfying $\int_0^xf=(f(x))^2+C$ for some constant $C \neq 0$. Find all continuous functions $f$ satisfying   $$\int_0^xf=(f(x))^2+C$$ for some constant $C \neq 0$, assuming that $f$ has at most one $0$.  I have a question about the solution to this problem. It says that   Clearly $f^2$ is differentiable everywhere, its derivative at $x$ is $f(x)$ (I'm assuming that this is because the left hand side is differentiable, the right hand side must be differentiable as well). So $f$ is differentiable at $x$ whenever $f(x) \neq 0$, and  $$f(x)=2f(x)f'(x),$$ so $f'(x) = \frac{1}{2}$ at such points. Thus, $f= \frac{1}{2}x+b$ for some $b$ on any interval where it is non-zero; if $f$ has a zero, this gives two possible solutions, with two possible values of $b$, but since $f$ is assumed continuous, they must be the same. So we need $$\int_0^x(\frac{1}{2}t+b)dt=(\frac{1}{2}x+b)^2+C$$  so we must have $b=\sqrt{-C}$ or $-\sqrt{-C}$, leading to two solutions for $C \lt 0$.  I don't understand the bolded statements. First, how does the differentiability of $f^2$ guarantee that $f$ is differentiable, and why only at $x$ where $f(x) \neq 0$?  Next, why are there two possible solutions if $f$ has a zero, but since $f$ is continuous they must be the same?  Finally, does the last statement mean that if $C \gt 0$, then there is no solution, since we're restricted to real numbers here?  I would appreciate it if anyone answers my questions. Would fewer Jedi be more powerful? Why does $\lim_{n\to\infty} z_n=A$ imply $\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A$? I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following.   Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$ \lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A. $$  Is there a clever way to write the limit to make it more approachable? Thank you. Suggested reading for renormalization (not only in QFT) Are token creatures considerd as a creature spell? Decode Base64 data in Java What's "Another Way In"? There's a new achievement and I'm stumped as to how to get it:     Start a game without breaking the laboratory wall   I assume it has something to do with the new Gadgets but none of them sound like they'd help. How do I get this achievement? A limit ordinal $\gamma&gt;0$ is said to be indecomposable iff $\nexists\alpha,\beta&lt;\gamma$ such that $\alpha+\beta=\gamma$.  In view of this definition, I’m trying to prove the equivalence of the following statements for limit ordinals $\gamma$:  (1) $\gamma$ is indecomposable.  (2) $\alpha+\gamma=\gamma$ $\forall\alpha&lt;\gamma$.  (3) $\gamma=\omega^\alpha$ for some $\alpha\in\mathrm{ON}$.  I’ve been able to get $(1)\implies(2)$ and $(3)\implies(1)$, but I have nothing more than scratch work for $(2)\implies(3)$.  Any suggestions would be appreciated. How to show  tag literally in / tag? Given (p ∧ q), use the Fitch system to prove (q ∨ r) I'm having trouble solving a Fitch Style Proof and I was hoping someone would be able to help me.  Premises: p ∧ q  Goal: q ∨ r  Would be an assumption (p ∧ q => q ∨ r) in first step a correct one?   Then, q ∨ r Implication Elimination: 2, 1  Why is it not correct?  Thank you very much for any help! limit find output AND avoid signal 13 I have a directory with ~1M files and need to search for particular patterns. I know how to do it for all the files:  find /path/ -exec grep -H -m 1 'pattern' \{\} \;   The full output is not desired (too slow). Several first hits are OK, so I tried to limit number of the lines:  find /path/ -exec grep -H -m 1 'pattern' \{\} \; | head -n 5   This results in 5 lines followed by  find: `grep' terminated by signal 13   and find continues to work. This is well explained . I tried quit action:  find /path/ -exec grep -H -m 1 'pattern' \{\} \; -quit   This outputs only the first match.  Is it possible to limit find output with specific number of results (like providing an argument to quit similar to head -n)? Determining the Actual Server Create Date I am interested in determining the ages of all 70+ my servers. I would like to create a growth graph showing when servers and databases were added to the environment either/both monthly or annually. I executed the following query using the SSMS multi-query feature with only partial success.  SELECT [name], create_date FROM sys.databases   It's partial success because the master and model databases return with a date of 'April 8, 2003' when I know that I created some of these servers during the past couple years.  I refresh databases from one server to another several times a week, which appears to be causing issues with the database create date.   Is there a simple method to determine the actual create date for both servers and databases? Creating Layer in GeoServer using REST As part of a possible solution to   I want create a layer in GeoServer via REST that supplies a SQL statement along with a userid. I think i would  and publish it. I need to do this all in REST. Does anyone have any code examples? What's the difference between STDIN and arguments passed to command? How to test real network throughput between two points? What are some of the better tools/utilities for testing real bandwidth across a link?  In my case I am testing the real throughput across a wifi bridge. Cannot redeem bounty I have about 25K of bounty vouchers but I cannot seem to redeem them. I believe I am sitting in the proper station and it has a office to redeem claimed bounties.  Bounty: 25,380 CR BOUNTY IN INDEPENDENT ERAVATE FREE  STATION: CLEVE HUB / INDEPENDENT ERAVATE NETWORK   If I am in the wrong station how do I find the information needed to travel to the correct station. Where can i see the faction running the station prior to flying there? for loop wget directory creation error As a Ubuntu beginner. I am trying to download links from multiple text files with wget2 in batch. The -P &quot;$f&quot; creates directory name with .txt suffix. Is it possible to ignore .txt for -P &quot;$f&quot;. Any modification suggestions. for f in *.txt;do wget2 -i &quot;$f&quot; -P &quot;$f&quot;;done Using Label Expression with Text Formatting Tags and Ampersand I'm trying to remember a cartoon tv show where the main boy, a blonde girl, and another boy capture monsters. There was something about a 'veil' between their world and the monster world, but a 'secret society' of sorts stopped the monsters from crossing, or used them to battle the people who wanted to use them for evil led by a wise old lady. I'm pretty sure they trapped monsters into pieces of jewelry or clothing.   Not so sure about this part: and they had these 'supernatural powers' that were connected to their 'aura' and they could travel from place to place with it.   (Another brown haired girl joined because she found out? There was an arc where the blonde got jealous and the monster she controlled attacked the brown haired girl?)  I can't remember much of it, but I remember in the end the 'veil' was broken and monsters flew over the city. And the main boy and the blonde girl held hands... or kissed? I downloaded Linux CAPT driver V2.60 from Canon and installed both .deb packages, which were listed under the 64bit folder with Ubuntu Software-Center. My Canon printer still won't be recognized. Is there anything else I have to do? My printer is connected over USB.  Ubuntu would make a great replacment for my office, but unfortunately without printer support I will still rely on Windows.