Here is the expression $$\sum_{k=0}^n\frac{1}{2^{k}}\binom{n+k}{n} = 2^{n}$$ How to go about proving this? I tried expanding the expression and it seems that $n+1$ must be even. But it isn't helping much. Can someone please help? Can this be proved using generating functions or combinatorics?
How does one show that $$\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$$ for each nonnegative integer $n$?  I tried using the  but I guess I am applying it incorrectly. With the snake oil technique we have $$F(x)= \sum_{n=0}^{\infty}\left\{\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}\right\}x^{n}.$$ I think I have to interchage the summation and do something. But I am not quite comfortable in interchanging the summation. Like after interchaging the summation will $$F(x)=\sum_{k=0}^{n}\sum_{n=0}^{\infty}\binom{n+k}{k}\frac{1}{2^k}x^{n}?$$ Even if I continue with this I am unable to get the correct answer.  How does one prove this using the Snake oil technique?  A combinatorial proof is also welcome, as are other kinds of proofs.
$A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$    If $\lambda$ is an eigenvalue of $A^2,$ we have $\lambda\alpha=A^2\alpha$ for some $\alpha.$ Then how can we find a $\beta$ s.t. $\sqrt{\lambda}\beta=A\beta?$ 16.04 Upgrade: Failed to fetch Empty files can't be valid archives Why is the Descriptor not getting called when defined as instance attribute? How to do the port forwarding from one ip to another ip in same network? I would like do some NAT in iptables. So that, all the packets coming to 192.168.12.87 and port 80 will be forwarded to 192.168.12.77 port 80.  How to do this with iptables?  Or   Any other ways to achieve the same? Verifying that the ideal $(x^3-y^2)$ is prime How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime?    I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel is exactly $I$. But the difficulty is that how to prove the kernel is contained in $I$.  Could you help me? Thanks a lot! If $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$ Let $a_1,a_2,a_3,\ldots$ be reals. Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$.   For this I have shown the case for when $ a_n^{2} \le\frac {|a_n|}{n}$ $\Rightarrow$ $ |a_n|\le\frac {1}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le \frac{1}{n^{2}}$ and we know that $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges and hence $\sum_{n=1}^{\infty}\frac {a_n}{n}$ converges by the comparison test. Now considering $ a_n^{2} \ge\frac {|a_n|}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le a_n^{2}$ $\rightarrow$ combining the two cases for any n we have: $\frac {|a_n|}{n}\le\frac{1}{n^{2}}+a_n^{2}$ Hence using the comparion test again we know that $\sum_{n=1}^{\infty} a_n^{2}$ converges and $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges hence the sum converges so we can conclude that $\sum_{n=1}^{\infty} \frac {a_n}{n}$ is absoluetly convergent $\Rightarrow$ convergent.  Not to sure if this is correct, any help would be much appreciated, many thanks. Should individual letters be preceded with "an"? I was reading a comic book and I noticed that the author used "an A" and "an N" in a particular phrase (screenshot below). While I understand the usage of "An A" here (A being a vowel and what not) and "N" having a pronunciation which has a sound similar to "A".  Since most of English alphabets have pronunciations which start with a syllable similar to one of the vowels or semi-vowels; is it OK to prefix the letters using "An"?  For eg. An X, or An Y.        Zenescope's Grimm Fairy Tales - Werewolves: The Hunger, page #8 (Click to enlarge) Is it possible to use the laptop's audio via headphone jack and Bluetooth speakers at the same time? Lolo Joe wants earphone jack to plug and use big speakers but at the same time he wants to partner the music coming out the earphone jack and his Bluetooth system and speakers. Is it even possible to use the earplug connection and the Bluetooth system and devices at the same time? How to convert decimal to hexadecimal in JavaScript How do you convert decimal values to their hexadecimal equivalent in JavaScript? "that which" used together When I read essays from Eliot, I find him using "that which" frequently, e.g.        the combination which is the murder of Agamemnon is probably as complex as that which is the voyage of Ulysses.   A very small part of acting is that which takes place on the stage!   They belong to a different race. Their crudity is that which was of the Roman, as compared with the Greek, in real life.      I can kind of guess its usage, but I want to know more about this grammar structure. Searching on Google mostly gave me the simple difference between "that" and "which", and some examples using "that which":        that which we call a rose (from "Romeo and Juliet")   that which we persist in doing      It is a pity that Google search does not direct me to any useful page about "that which".  Can someone explicate its grammar for me? delayed addclass/remove class function not working Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Proving that the set of $\nu$ measurable sets $M\subseteq P(X)$ is an algebra How do you format the day of the month to say "11th", "21st" or "23rd" (ordinal indicator)? Not getting edit's +2 rep after reaching 2k rep After reaching 2k rep , I'm not receiving  +2 rep per edit. Why? Is $\int_{-\infty}^{\infty} \sin x \, \mathrm{dx}$ divergent or convergent? I was determining whether   $$\int_{-\infty}^{\infty} \sin x \, \mathrm{dx}$$  was divergent or convergent. So, I did the following steps:  $$\begin{align} \int_{-\infty}^{\infty} \sin x \, \mathrm{dx} &amp;= \int_{0}^{\infty}\sin x \, \mathrm{dx}+\int_{-\infty}^0\sin x \, \mathrm{dx} \\ &amp;=\lim_{t\rightarrow\infty} \left(-\cos x |^{t}_{0}\right) + \lim_{a\rightarrow-\infty} \left(-\cos x |^{0}_{a}\right)\\ &amp;=\lim_{t\rightarrow\infty} -\cos (t) + \cos 0 + \lim_{a\rightarrow-\infty} -\cos 0 + \cos a\\ &amp;=\lim_{t\rightarrow\infty}1 - \cos t + \lim_{a\rightarrow-\infty} -1+\cos a \end{align}$$  Now, at this point, it would be reasonable to say that both the limits are undefined and therefore, the integral is divergent but then if I try something like the following   \begin{align} \quad\quad&amp;=\lim_{t\rightarrow\infty}1 -\cos t + \lim_{a\rightarrow\infty} -1+\cos a \\ \quad\quad&amp;=\lim_{t\rightarrow\infty}-1 -\cos t + \lim_{a\rightarrow\infty} \cos a+1 \\ \quad\quad&amp;=\lim_{b\rightarrow\infty}-1 +\cos b - \cos b+1 \\ \quad\quad&amp;= 0 \end{align}  So, as you can see, it was shown before that the integral is divergent but with some manipulation, we came at an answer of $0$ but is that valid? I assume, a similar technique can be applied to $\int_{-\infty}^{\infty} \frac{1}{x} \, \mathrm{dx}$. Sheet Specific Scripts I'm currently using the below script to timestamp when an edit is made to column 7 on a sheet named "Sheet 1".  function onEdit(e) {    if ([7].indexOf(e.range.columnStart) != -1) {    e.range.offset(0, 7).setValue(new Date()).setNumberFormat("dd.MM.yyyy at HH:mm");    }   }     I have a second sheet "Sheet 2" which I need to run the same script on but against column 5.  function onEdit(e) {    if ([5].indexOf(e.range.columnStart) != -1) {    e.range.offset(0, 7).setValue(new Date()).setNumberFormat("dd.MM.yyyy at HH:mm");    }   }     I've tried .getSheetByName("Sheet1") but just cant seem to get it working correctly. Replacing Raster Values in Ras1 with Ras2 How does TikZ sort intersections? I used the sort by key for intersections to sort along a line, expecting the intersections to be numbered 1 through 5. Why were they numbered 4, 5, 3, 1, 2? Playing around with it, I could not seem to figure out how the order is determined. The code and image follow.  \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{intersections}  \begin{document} \begin{tikzpicture}     \draw[name path=grid] [xstep=3,ystep=2] (0,0) grid (9,8);     \draw[-&gt;, name path=line] (2,1) -- (7,7);     \draw[name intersections={of=grid and line, sort by=line, name=i, total=\t}]         \foreach \s in {1,...,\t}{(i-\s) node {\s}}; \end{tikzpicture} \end{document} Error message: qgis_core.dll missing I have a fresh install of QGIS 2.4. When trying to open a .shp file I get the error message below.  I've followed their advice and have done several re-installs and also cleared all qgis files from my system (Parallels on Mac OS X 10.6) before those re-installs. I've tried QGIS 2.2 which caused the same problem.  Funnily all worked for a day, the following day and since not anymore...  Has anyone come across this before?