$$\begin{cases} {a_n} = 3a_{n-1} - 3a_{n-2} + a_{n-3}\\ {a_0} = 1, a_1=2, a_2=5 \end{cases}$$  Hey everyone..  So far I have solved only questions of 2 root kind. I tried to solve this one but no success.. I dont know what do to after i found out:  $$x^{3} - 3x^{2} + 3{x} - 1 = 0$$ equals to: $$(x-1)^{3}$$  Aaaand lost.
Suppose $a_0,a_1,a_2$ satisfy the recurrence $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}$ for $n\ge3$   Let $c_n=a_{n+1}-a_n$ for $n\ge1$ and $c_0=0$   Find a linear recurrence of degree at most 2 for the sequence $c_0,c_1,c_2,...$  I'm not entirely sure what they are asking for but I think it is asking for a recurrence relation for $c_n$ where the characteristic polynomial has degree at most 2.  I tried finding a closed form for $a_n$ to try and get rid of the $a_n$ in the $c_n$ equation, but I can't figure it out when the initial conditions aren't specified ($a_0,a_1,a_2$)  Any help would be appreciated!
Exclude TeX.SE question titles from MathJax parsing in Hot Network Questions Recently, the TeX.SE question  got onto the Hot Network Questions list.  Like many questions on TeX.SE, it contains some TeX markup in its title.  Unfortunately, on , this markup gets parsed and rendered as math:    I propose that either:   the entire Hot Network Questions sidebar should be excluded from MathJax parsing by styling it with the tex2jax_ignore class, or at least, any hot questions from TeX.SE should be so excluded from MathJax parsing.   Or I suppose we could always implement , which asks for all questions with TeX in the title to be excluded from Hot Network Questions entirely.  That might be throwing out the baby with the bathwater, though.    Ps. This issue is partially masked by , which causes MathJax not to be rendered in many elements that are hidden by default and revealed using jQuery .show(), such as all but the first five questions in the Hot Network Questions sidebar.  The  user script contains a generic fix for that bug, so users who have it installed will see this issue more frequently, but it occurs even without SOUP whenever a question with TeX markup in the title happens to show up among the top five hot questions that are always visible by default. Google Chrome on a fresh CentOS 7.1 not starting up These days, the standard way to present differential calculus is by introducing the Cauchy-Weierstrass definition of the limit. One then defines the derivative as a limit, proves results like the Leibniz and chain rules, and uses this machinery to differentiate some simple functions such as polynomials. The purpose of my question is to see what creative alternatives people can describe to this approach. The nature of the question is that there is not going to be a single best answer. I have several methods that I've collected which I'll put in as answers to my own question.  It's not reasonable to expect answers to include an entire introductory textbook treatment of differentiation, nor would anyone want to read answers that were that lengthy. A sketch is fine. Lack of rigor is fine. Well known notation and terminology can be assumed. It would be nice to develop things to the point where one can differentiate a polynomial, since that would help to illustrate how your method works and demonstrate that it's usable. For this purpose, it suffices to prove that if $n&gt;0$ is an integer, the derivative of $x^n$ equals $0$ at $0$ and equals $n$ at $1$; the result at other nonzero values of $x$ follows by scaling. Doing this for $n=2$ is fine if the generalization to $n&gt;2$ is obvious. Why do we use "./" (dot slash) to execute a file in Linux/UNIX? Why do we use ./filename to execute a file in linux?  Why not just enter it like other commands gcc, ls etc... bi-directional sync with rsync Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$ Let $f(z)$ be an entire function s.t $|f(z)|\leq M|z|$ ; $M&gt;0$ is a constant then $f(z)$ Is polynomial of degree 1   My try $f(z)$ is entire so it can be written as a $\sum a_n z^n$  $$|\sum a_n z^n|\leq \sum |a_n||z|^n\leq M|z|$$  So $n=1$  Seems that the move $\sum |a_n||z|^n\leq M|z|$ is problematic Related:   I know I must describe my past and current research in my personal/research statement or statement of purpose. But do I use technical terms?  I was given the advice that since the people reading my statement are mathematicians (I am applying to US for applied mathematics), they should know what I am talking about.  Although I am applying to applied mathematics, it would also be nice to hear from other fields (biology, computer science, engineering etc) about this. How do you convert Html to plain text? I have snippets of Html stored in a table. Not entire pages, no  tags or the like, just basic formatting.  I would like to be able to display that Html as text only, no formatting, on a given page (actually just the first 30 - 50 characters but that's the easy bit).  How do I place the "text" within that Html into a string as straight text?  So this piece of code.  &lt;b&gt;Hello World.&lt;/b&gt;&lt;br/&gt;&lt;p&gt;&lt;i&gt;Is there anyone out there?&lt;/i&gt;&lt;p&gt;   Becomes:  Hello World. Is there anyone out there? How do I remove Windows but keep Ubuntu? I want to keep Ubuntu but remove Windows. This works:   This does not (blank image):   Notice the 1.1.1 -> 1.3.0 and SRS->CRS are the only changes.  Is this a problem with the service or is my second URL incorrect?  I've observed the same issue with other WMS services:  Works:   Does not work (blank image):   So I'm assuming it's an issue with my 1.3.0 URLs. How can I diagnose and repair missing drive space? I have a small ext3 / partition on a 2T drive which appears to have a discrepancy in its free space. How can I determine what is causing it and how it can be fixed.   After trying everything I could think of, I forced a fsck on boot which didn't have any affect.  I have other systems configured identically which don't exhibit this condition.  # df -h / Filesystem            Size  Used Avail Use% Mounted on /dev/sda1             9.7G  8.9G  318M  97% /  # du -shx --max-depth 1 / 5.2G    /  # lsof | awk 'BEGIN {t=0} /(deleted)/ {t+=$7} END {print t}' 0  # tune4fs -l /dev/sda1 | grep -E 'state|Free|Reserve|size|Inode' Filesystem features:      has_journal ext_attr resize_inode dir_index filetype needs_recovery sparse_super large_file Filesystem state:         clean Inode count:              2621440 Reserved block count:     131029 Free blocks:              212317 Free inodes:              2487281 Block size:               4096 Fragment size:            4096 Reserved GDT blocks:      639 Inodes per group:         32768 Inode blocks per group:   1024 Reserved blocks uid:      0 (user root) Reserved blocks gid:      0 (group root) Inode size:               128   Edit:  fsck says the partition is 2% non-contiguous.  For comparison to du above (sparseness):  # du -hxs --apparent-size / 4.9G    / How do you reverse a string in place in C or C++? How do you reverse a string in C or C++ without requiring a separate buffer to hold the reversed string? Show $1/(1+ x^2)$ is uniformly continuous on $\Bbb R$. What word indicates spatial data within the geographical and geological domains? In particular, what word is precise enough for a science journal article (i.e., formal prose)? Candidates might include &quot;spatial data,&quot; &quot;geodata,&quot; &quot;geospatial data,&quot; and &quot;geographic data.&quot; &quot;Spatial data&quot; seems too generic (not necessarily within the geographical and geological domains). What is the consensus, if any exists? When reviewing an answer in the low quality queue the question doesn't believe that I have enough reputation to comment.        I can't remember if commenting on the post not actually being reviewed was ever allowed? What kind of questions can I expect to see on a qualifying exam? Why does Worf want O'Brien to go to Minsk? Enumerate after \label Using Unicode Zero Width Spaces allows all space comments Two Bibliographies: one for main text and one for appendix I have a manuscript in which has the following structure:  Main Text Bibliography Appendix   However, there are citations which are only cited in the appendix, and they show up in the main bibliography. I would like to split my Bibliography into two different sections, so that the structure of the paper would look like this:  Main Text Bibliography for Main Text Appendix Bibliography for Appendix   I'm currently using bibtex and a single .bib file, and creating the bibliography with  \bibliographystyle {someBibStyleFile} \bibliography {bibFileName}   I'm interested in a method to split my single bibliography into main bibliography and appendix bibliography with minimal changes to the rest of the document.  I'm aware that biblatex is more powerful than bibtex, but I'm not prepared to make the switch before the deadline of this project. There are many related bibliography-with-multiple-sections type of questions, but I didn't see one that covers this case.