Proving a combinatoric proof using the result of $\sum_{k=0}^n \binom nk\ (-1)^k = 0$
Help ending a proof using binomial theorem: $\sum_{k=0}^n {n \choose k}\cdot\frac{(-1)^k}{k+1} = \frac{1}{n+1}$
What theorems can I use to prove this? line segment joining two centers of circles is perpendicular to line segment joining two intersection point of the circles I am stuck on the following question:     Find the volume of the torus obtained by rotating the circle $(x-a)^2+y^2=b^2$, where $a&gt;b$, around the y-axis. Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? Why does $\lim\limits_{t\to-\infty}te^t=0$? Intuitively, $\lim\limits_{t\to -\infty}te^t=0$, since $e^t\to 0$ much faster than $t\to -\infty$.   Is there a way to more rigorously compute this? Writing it as $e^t/(1/t)$ seems resistant to l'Hopital's rule. Insert list of figures in the table of contents Is it possible to create a list of all the figures at the end of a document (this I can do) and make it appear in the table of contents?  So in the the table of content there will be a line saying List of figures ... 19 for example? Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$ Internal or external oscillator I always use the internal oscillator that pics have as I have never found the need to run anything at higher frequency than 8 MHz (which is the fastest the pics I use tend to be able to go). Are there any reasons, beyond going above 8 MHz, that mean I should use an external oscillator? It just seems like one more thing to go wrong to me but I'd be interested to hear what others do. What counts as a 'large' raid 5 array? Can I use Named Credential merge fields as URL parameters on the endpoint? How can I find more music similar to the ones I like? How can I configure the Ubuntu Dock to show windows only from the current workspace? I would like my Dock to show only the windows of the current workspace. Especially in the little preview window. I find it very confusing to jump around workspaces when trying to find the right window. Complex polynomial and the unit circle Given a polynomial $ P(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0 $, such that   $\max_{|z|=1} |P(z)| = 1 $      Prove: $ P(z) = z^n $      Hint: Use cauchy derivative estimation    $$ |f^{(n)} (z_0)| \leq \frac{n!}{r^n} \max_{|z-z_0|\leq r} |f(z)| $$ and look at the function $ \frac{P(z)}{z^n} $     It seems to be maximum principle related, but I can't see how to use it and I can't understand how to use the hint. I want to encrypt a USB stick, and be able to decrypt it on Windows as well as Ubuntu. How do I do this &amp; what programs do I need to use? Salesforce1 on focus scroll to the top Greek pdf bookmarks I am puzzled with sections titled with Greek characters. They do appear in the pdf's Content list, though not correctly.  For example, in my thesis--written in English with LyX, based on the KOMA-script--I choose to include a translation of the summary in Greek (=Περίληψη). The "raw" code for the (unnumbered) chapter title reads: \addchap{\textgreek{Per\char208lhyh}}. In the final .pdf file (using pdflatex) the respective entry appears as Per208lhyh. Using the unicode option for the hyperref package does not fix the problem.  Update detail: the document (should) not only contain Chapter or Section titles in Greek, but normal text as well.  I have found some relevant references within tex.stackexchange and elsewhere (e.g., , the manual of the textgreek package, , ). However, I cannot extract a concrete answer.  How can Greek characters appear correctly as bookmarks in the .pdf's Content list?  Note, the information of this Q&amp;A applies to/under LyX as well. How to add custom C++ keywords to be recognized by Minted? When doing an apt-get update/upgrade I see that warning  W: Conflicting distribution: https://downloads.plex.tv/repo/deb public InRelease (expected public but got )   I was upgrading to Ubuntu Zesty 17.04 before and do not know how to fix the above. Already have seen ? but there are no suggested fixes for that.  Maybe one of you can give some hints? Thanks! Proof that $\inf A = -\sup(-A)$ Is my processor 64 or 32 bit?