Find the limit $$\lim_{n \to \infty}\left[\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\right]$$   I take log and get $$\lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)$$
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 $1$?
When is a gerund supposed to be preceded by a possessive adjective/determiner? I assume that the following sentences are grammatically correct:   He resents your being more popular than he is. Most of the members paid their dues without my asking them. They objected to the youngest girl's being given the command position. What do you think about his buying such an expensive car? We were all sorry about Jane's losing her parents like that.   I'm still getting used to this possessive gerund structure. It sounded so weird to me at first. Is the structure used in both formal and informal contexts? Are there any alternative structures that result in the same meaning and are more frequently used? (Examples taken from ) Google Chrome on a fresh CentOS 7.1 not starting up Is there something wrong with VertexConnectivity How does a magnet work? I'm having trouble understanding how a  (not the  that is generated as a result but the material itself) work. The particles are aligned in a specific direction to give rise to force but I don't see how this alignment gives rise to "attraction" or force. Is there a label/goto in Python? Is there a goto or any equivalent in Python to be able to jump to a specific line of code? Syntax highlighting in app differs from normal web view Differences between Verb + to be + adjectives and Verb + adjective Easiest way to move my Windows installation to an SSD? I've taken the plunge and bought an SSD and want to move my existing Windows installation over. The current hard disk is 500Gb, but I've trimmed the contents down to about ~40Gb. I'm transferring it across to a 100Gb SSD and looking for the easiest way just to copy everything across and set the SSD up as a boot device.  I've looked at a few tools like Macrium Reflect, but they don't seem able to restore to a smaller drive. Do I need to go for something like  to do this? I'm trying to avoid scary Linux-based boot utilities if possible, does anyone know of an easier way? I've been messing around with Unity and broke something, how do I "start over"? I wrote a question about blocking certain characters in passwords. It became tremendously unpopular, and it seemed like everyone wanted to rain on my parade every day.  Eventually, I "got it," agreed completely with what people were saying to me, and just wanted to withdraw my question so that my reputation didn't slide even further downhill.  Unfortunately, the system is designed such that I can't withdraw my question or redeem myself. Instead, I just get daily reputation attacks. That's a terrible system, and it needs to be changed. Fast way to come up with solutions to $x(x-1)(x-2)(x-3)=1$? I can solve this equation $x(x-1)(x-2)(x-3)=1$ using the usual method but I am looking for a fast analytical method to solve this. Any hints ? Prove no existing a smooth function satisfying ... related to Morse Theory Are the proofs for the properties of parallel lines, and that a triangle has 180 degrees, inherently tautological? I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.   So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.   Thanks in advance. Duplicate objects transformation applies to both objects I'm facing this frustrating "feature" that I can't seem to remedy; when I duplicate an object (and now, even when I create two completely new objects), their transformations/scaling is synchronized. This is extremely annoying when trying to use any sort of transformation keyboard shortcut.  It should be noted by going to the properties menu (N) and manually setting the properties works just fine.  For instance, here is just the camera selected (trust me, I made sure):   And when I try to transform by clicking+dragging on the X axis OR by hitting G->X, this happens:   They are not grouped (verified by going to the Groups list up in the Scene panel) and they do not have parents (verified by going to Object->Relations).  I have tried the unlink command as well, to no avail.  I have seriously run out of options; what did I change to have my G->X shortcuts transform multiple objects? It seems completely at random, too; certain objects seemed to be "linked" to other random objects, and this "linking" is guaranteed when duplicating even when I do a non-linking duplicate (Shift+D).  What gives? Can a plan "age" reach zero? Book a return flight a year in advance Is it possible to buy an airline ticket that has me returning to my country after a year being abroad? I can't seem to book a flight that far in advance online. What is the relation between normal extension and separable extension? What is the relation between normal extension and separable extension? Let F be the algebraic extension of K, if F is a separable extension of K,if and only if F is a normal extension of K? is this correct? I am a bit confused. When can we omit the preposition on before weekdays? (Monday, Tuesday etc.) Sometimes I read weekdays without the preposition on. If the preposition is left out, does it work the same way, or does it change the meaning?  Example without the preposition     The UGC issued the public notice Monday after its two previous orders   in this regard were overlooked by the DU.   Example with the preposition     The commission on Monday also issued a public notice against the FYUP   in all leading newspapers for parents and students. While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. I was trying to install Teamviewer, but I was getting a dependency error. I tried to install the required packages, but with no luck.  I'm getting this error:  Unpacking teamviewer (from teamviewer_linux_x64.deb) ... dpkg: dependency problems prevent configuration of teamviewer:  teamviewer depends on lib32asound2; however:   Package lib32asound2 is not installed.  teamviewer depends on lib32z1; however:   Package lib32z1 is not installed.  teamviewer depends on ia32-libs; however:   Package ia32-libs is not installed.  dpkg: error processing teamviewer (--install):  dependency problems - leaving unconfigured Errors were encountered while processing:  teamviewer   I tried   sudo apt-get -f install   Getting   Package ia32-libs is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source However the following packages replace it:   lib32z1 lib32ncurses5 lib32bz2-1.0  Package lib32asound2 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'lib32asound2' has no installation candidate E: Package 'ia32-libs' has no installation candidate   I can't even get to the   sudo dpkg -i teamviewer_linux_x64.deb   If I force installation  sudo dpkg --force-depends -i teamviewer_linux_x64.deb   Although it's "Setting up Temviewer" it gives me this:    How can I fix this problem?