I have to find the power series of $$\frac{1}{1+x^{2}}$$ centered at $a=1$. So, I tried to write it as $$\frac{1}{1-(-(x-1)^{2})}$$ to sue the fact that $$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$ Since $1-(-(x-1)^{2})=1+x^{2}-2x+1$, then what I did next was: $$\frac{1}{1+x^{2}}=\frac{1}{1+x^{2}-2x+1+2x-1}=\frac{1}{1+(x-1)^{2}+2x-1}=\frac{1}{2x+(x-1)^{2}}$$ My question is: what am I supposed to do after this? I also thought about doing something like: $$\frac{1}{1+x^{2}}=\frac{1}{1+x^{2}-2x+1+2x-1}=\frac{1}{1+(x-1)^{2}+2x-1}=\frac{1}{(1+(x-1)^{2})+(2x-1)}$$ and then, using partial fractions, try to finish it.
I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$  I'm not allowed to use taylor expansion. So my first thought was to rewrite the function in a form where I can apply the basic identity:  $$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n $$  So let's rewrite the original function $f(z) = \frac {1}{1+z^2} $ and use above identity:  $$\frac {1}{1+z^2} = \frac {1}{1-(-z^2)} $$ But observe that I need to center it at $z_0=1$ so I rewrote it to:  $$\frac {1}{1-(-z^2)}=\frac {1}{2z-(-(z-1)^2)} $$  I can factor out $\frac {1}{2z}$ and use the basic identity to get : $$\frac {1}{2z}\frac {1}{1-\frac{(z-1)^2}{2z}} = \frac {1}{2z}\sum_{n=0}^\infty \left(\frac{-(z-1)^2}{2z}\right)^n =\frac {1}{2z}\sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^n} = \sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^{n+1}}$$  The problem is if I ask wolfram alpha: power series$ \frac{1}{1+z^2} $centered at z = 1 it will give me a different answer that doesn't match the sum I found.  Can anyone help me figure this one out? What am I doing wrong?
If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism. Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism.   My attempt:  We need only show that $T^{-1}$ is continuous.  Let $M \subset Y$ be closed. Then, from the surjectiveness of $T$, there exists a $A \subset X$ such that $T(A)=M$. Since $M$ is closed, by the continuity of $T$, we must have that $A$ is closed.  Now $$T^{-1}(T(A))=T^{-1}(M)$$ i.e, $A=T^{-1}(M)$. Since $A$ and $M$ are closed, we must have that $T^{-1}$ is continuous.  My problem with this, however, is that I am not using the fact that $X$ is compact anywhere in the proof. Is there perhaps another way that I can show this, using the compactness of $X$?   Any hints will be appreciated. Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares I'd appreciate your help showing that if $p\equiv3\,\left(\mbox{mod 4}\right)$   then p can't be written as a sum of two squares.  Thanks! How to start a terminal with certain text already input on the command-line? Rather than rephrasing my question, let me describe to you the desired user-case:  I create a short shell-script to run command "gnome-terminal --someoptionflagname 'my text to be posted'", and execute this script.  Gnome-terminal pops up, with command-line prompt followed by my text.   ie: fields@mycomputer:/$ my text to be posted  Can this be done? Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” Why does everyone do 9-overlord, not 10-overlord? What is the advantage on saving the money for the overlord when you have 9 drones, and not filling your supply first and then getting the overlord?  In my theory, in the first case you get to wait the saving time with 9 drones, and get to wait almost the same time with 10 drones in the second case. Second case should be better, since 10 drones mine faster than 9? So why does everybody go with the first case?  All this is assuming you go 12 pool or something like that, where you will get more drones after the overlord. 6- and 8-pools are obviously different. Windows 10 update broke my MBR / GRUB Fisher's exact test on 2x4 table I am working on 4 different plants. I have the (RNA-Seq) data from sequencing. I look for two events E1 and E2, say, at certain positions in their genome. Let's say E2 is the common one and E1 is a special event. The positions I look for are identical across all 4 (in the genome). And let's say I see that the observation I have for 1 particular position is:      P1   P2  P3  P4 E1   0   20   0   17 E2 100   80 100  120    Here, P1 through P4 refers to plants and E1 and E2 refers to the events. E2 is more common. So, my objective is to actually check if E1 occurs more often in one or more plants than in the others. If they occur at the same proportion in all, of course it is not interesting to me.  I have 2 questions: (I have already asked  before but didn't get an answer)     Will a fisher test be right for this problem?    I hypothesize (Null) that the proportion of E1 is not different in occurrence between the 4 plants. Now, I set out to find if the proportion I have here is by any means significantly different. I use R, fisher.test() and I get p-value=2.5e-10. So, I reject my Null hypothesis for this case because I find strong evidence against it.  Sometimes, I have an observation like this,        P1   P2  P3  P4 E1   0   20   0  17 E2   0   80   0 120    then fisher.test() gives me a p-value of 0.147. So, I don't reject the Null   hypothesis. However, from a biological point of view, I would consider this significant. However fisher test answers the question I originally asked. I guess the proportion 0/0 for P1 and P3 are not useful (or not used).  So, my question is: Is it possible to modify the test such that it is sensitive even if E1 and E2 are 0 in 1 or more plants for a particular observation?     Having thought a bit to frame this post, I guess, in that case I have to ask a different question. Moderator flag notifications should reach mobile I spend a great deal of time surfing SE sites from my phone during commutes and other down times. The global inbox is quite effective at reaching me even for chat pings. Quite frankly the mobile site is very well done!  However as a community moderator, there is a whole class of information not getting through. There are no notifications for moderator flags! I can switch to the full site and see these but that is inconvenient for browsing, so I only end up checking for flags periodically.  Ideally the moderator tools should all be available in mobile templates, but as a stand-by it's reasonable to switch to the full view to handle flags, but flags would get addressed in a much more timely manner if I knew about them. As it stands I often come and go from the site without even knowing there are outstanding flags.  A simple flag counter added to the header next to the notification counter and linked to the tools page (which automatically opens the full version anyway right now) would fix this. Show that if $n>2$, then $(n!)^2>n^n$. Show that if $n&gt;2$, then $(n!)^2&gt;n^n$.     My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n&gt;(n+1)^{n-1}$ Here, I tried to use induction again without any luck. I also took log of both sides, but I did not get anything. Please help! Logarithm as limit lists this as a limit representation of the natural log:  $$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$  Is there a quick proof of this? Thanks Calling sleep(5); and updating text field not working Show that, in any finite graph with at least two vertices, there must be two vertices with the same degree.  HINTS ONLY! Can't get armature to move mesh! In the past month, I've been really busy with the university and stuff, and I don't sleep as much as I used to. I usually sleep 4-5 hours a night.  I haven't been doing that extensively ( I always kept a good sleeping routine), but can the lack of sleep affect my health on the long term ? Using latexdiff with git Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from. Wulfgar won't teach me Unrelenting Force's final word after Arngeir accepts the Horn of Jurgen Windcaller Why is this power adapter transformerless? Why am I getting a NoClassDefFoundError in Java? This one never ceases to confuse me. When to use 'such as' and when to use 'like' while giving examples? Is there any clear rule?        Metros like Mumbai, Delhi and Karachi are unsafe after dark.    Metros such as Mumbai, Delhi and Karachi are unsafe after dark.       (Before anyone points out, Delhi and Karachi are unsafe round-the-clock, but it's just an example.)