How could I calculate this sum:  $$\sum\limits_{k=0}^{\lfloor \frac{n}{2}\rfloor} {n \choose 2k}$$
What is the closed formula for finding      $$ \sum_{i=0}^{k} \binom{n}{2i}=?\quad ,k\leq \lfloor{n/2}\rfloor $$   I need summation only up to some intermediate  $k$ where $k\leq \lfloor{n/2}\rfloor$ .  SAY I need a closed formula to calculate $\binom{8}{0}+ \binom{8}{2}+\binom{8}{4} +\binom{8}{6}$  HERE $n=8$ and $k=6$
Humans don't digest fibre, and nor do any animals, but some creatures have micro-organisms (bacteria, I'm guessing) in their gut which digest fibre for them.  My question is, could humans take some sort of pill and establish our own fibre-digesting colonies in our intestines?  Has this been tried?  Supposing it could be done, would you suppose it would have any negative effects on our health?  (i.e.. would those whose gut digested fibre for them be stuck on a low-fibre diet?).    Purpose of question: just wondering.  It would be cool to be able to eat bark. Understanding 74LVC245 datasheet and current source/sink I'm using a SN74LVC245A () to shift 5V TTL down to 3.3V so I can read the signals with a Raspberry Pi.  However I am getting really weird results (a 60Hz input signal is picked up by the Pi at anything between 110Hz and 320Hz) and I'm really confused about how to read the datasheet to figure out whether I need pullup or pulldown resistors.  The datasheet says on page 4 the "high-level output current" is -24mA, and the "low-level output current" is 24mA.  So to me this means that when the device is outputting a high signal, it is able to sink 24mA of current (since outputting a negative amount of current suggests the current is going into the device instead.)  But then if you're outputting a positive voltage, isn't that sourcing current?  Why is the figure listed as a negative value?  Is anyone able to explain what these figures actually mean?  When I disconnect the 74LVC245 output from my GPIO pin, I still get weird signals from the floating input pin so I am assuming this means the 74LVC245 output is also floating in one state, but I don't know in which state it is (i.e. is it left floating on a high output signal or a low one?)  It doesn't really make sense to be floating on a high output signal since it's a level converter and the whole point is to maintain the correct 'high' voltage, so do I need pulldown resistors to handle the low output?  If so, why does the datasheet seem to suggest the chip can both source and sink current?  (In my mind, source current means the device can output a current with a positive voltage, while sinking current means the device takes in anything and connects it to GND, so perhaps that is where I am going wrong.)  Anyway I'm quite confused so any clarification would be much appreciated! I have noticed that guest quarters on the Enterprise D have fruit bowls on their tables.  A prominent instance is the fruit bowl in Berlinghoff Rasmussen's quarters in "".  The first thing that Rasmussen does when Data takes him to his guest quarters is take a piece of fruit and shove it in his pocket.  Who replicates fruit for these fruit bowls before guests arrive?  More generally, who removes unused fruit and tidies quarters after guests leave?  Is there a ship hospitality staff, perhaps including Guinan's wait staff in Ten Forward? How to limit current of an AC signal For my project, I'm trying to work with the principle of electrovibration. This principle is about simulating textures using low current electricity on an insulated conductive surface ().  In order to do this, I will be experimenting with generated AC signals of 80 to 160 Vpp at 60Hz to 400Hz. I will do this by using an DC - AC converter, usually used for EL lamp applications.  My only concern is how to limit the current coming from the converter. Are the techniques for limiting AC current the same as DC, by using a simple resistor? Or is it, as I expect, more complex?  I need to limit the current to 0.5mA, for safety reasons. Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise. I have the following page (deadlink: http://www.workingstorage.com/Sample.htm ) that has a footer which I can't make sit at the bottom of the page. I want the footer to  stick to the window bottom when the page is short and the screen is not filled, and stay at the document end and move down as normal when there is more than a screenful of content (instead of overlapping the content).  The CSS is inherited and befuddles me. I can't seem to change it properly to put a minimum height on the content or make the footer go to the bottom. Does a particle annihilate only with its antiparticle? If yes, why? Or to put the question another way - what is the result of a proton-positron collision, or an up quark-charm antiquark collision, etc.? As far as I know, annihilation happens only between particles of opposite charge and same mass, but perhaps I am wrong?  And if the types of annihilation mentioned above cannot happen, what are the reasons? Responding to your "too localized" concerns Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)         For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\bf{\frac{a_m}{a_n}}$   There exist $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$      My attempt:  Analysing question in light of $a_k=k\implies x_k=\frac{k+1-k}{k+1}=\frac1{k+1}$, I feel both are correct, I doubt if  I can generalise the result in light of this one example.   Any way to prove that this true for general case?    Note: Please do not confuse  with my question. The first two options are different, I've marked the difference in bold. Are 7 hours enough time to leave the Brussels airport and go to the city center? Use induction to prove the following: $1! + 2! + .... + n! &lt; (n + 1)!$  Base case:  $n = 1$  $1! &lt; 2!$ true   Inductive step: Assume that $1! + 2! + .... + k! \le  (k + 1)!$ is true   let $n = k + 1$  $1! + 2! + .... + (k + 1)! &lt; (k + 2)!$  $1! + 2! + .... + k! + (k + 1)! &lt; (k + 2)!$  $1! + 2! + .... + k! + (k + 1)! &lt; (k + 2)!$  $(k + 1)! + (k + 1)! &lt; (k + 2)!$  $2(k + 1)! &lt; (k + 2)!$  $2(k + 1)(k)(k - 1).. &lt; (k + 2)(k + 1)(k)(k - 1)....$  $2 &lt; (k + 2)$   the right is always bigger for k are non negative integer. Zero-dimensional ideals and finite-dimensional algebras Session_ID_Remove Issue in API callouts in Salesforce How to get apt-cacher-ng to download AND cache packages from Apt HTTPS repositories? Can't delete files in trash because they don't exist on OS X 10.6? Something funky happened on one of my external drives and now whenever I plug it in 2 files show up in my Trash.  If I empty the Trash via Finder, nothing happens.  Ditto for the Option+Empty Trash method.  Listing them via Terminal shows them as there  $ sudo ls .Trashes/501 ¶_|íìê&lt;-.b?†            æâù©ìë—‡.*?§   But attempting to rm them or get any information about them shows them as not there  $ sudo ls -F .Trashes/501/ ls: ¶_|íìê&lt;-.b\032†: No such file or directory ls: æâù©ìë—‡.*\026§: No such file or directory  $ sudo rm .Trashes/501/* rm: .Trashes/501/¶_|íìê&lt;-.b\032†: No such file or directory rm: .Trashes/501/æâù©ìë—‡.*\026§: No such file or directory   Any thoughts?  I've never seen anything like this.  Thanks in advance! What is the difference between a population and a sample? What is the difference between a population and a sample? What common variables and statistics are used for each one, and how do those relate to each other? How to get week number in Python? Encouraging users to create an account (and keep it) states that the isomorphism type of a finite simple group is determined by its order, except that:   $L_4(2)$ and $L_3(4)$ both have order $20160$ $O_{2n+1}(q)$ and $S_{2n}(q)$ have the same order for $q$ odd, $n &gt; 2$   I think this means that for each integer $g$, there are $0$, $1$ or $2$ simple groups of order $g$.  Do we need the full strength of the  to prove this, or is there a simpler way of proving it? Is $\mathbb{Z}[x]/(x^2+1)$  isomorphic to  $\mathbb{Z}[i]$?   My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. If it is possible then $\ker g$ is $(x^2+1)$? Am I on the right track? Please Help.