I am trying to evaluate a sum involving binomial coefficients, and by some manipulations, I have reduced it to $$\sum_{i=0}^{n} \binom{n+i}{i} 2^{n-i}$$ where $n$ is a constant ($n=1009$ in my particular case). (This looks like the LHS of the Hockey Stick Identity, except for the presence of the $2^{n-i}$ term. I would also be interested in further generalisation, replacing the $2$ by an arbitrary constant).  To evaluate this, I attempted writing $$2^{n-i} = \sum_{k=0}^{n-i} \binom{n-i}{k}$$ which gave a more symmetric expression, but otherwise didn't seem to help much.  So, given that the answer (according to Mathematica) is $4^n$, how can this be proved?
I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$  I know that $$2^n = \sum\limits_{k=0}^n \binom{n}{k}$$ but that didn't help me
I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number and wondered how normal this was. The following is what I've done in my investigation so far.  In the first figure below, I have plotted the probability that some number below $10^k$ has $m$ indistinct prime factors.  The second figure shows the evolution of the probability of finding a number with $m$ indistinct prime factors as the numbers one is looking at grows, i.e. it is just another way to visualize the same data. I have left out all points of zero probability.  If one looks at the first part of the figure, it seems that the peak of the curve for some $k$ shifts toward higher $m$ as $k$ grows.   My first question is this: Does this behavior (i.e. the peak shifting toward higher $m$ as $k$ grows) continue indefinitely, or will the peak stabilize at some point? The latter seems unlikely but I can't be sure.  My second question is this: Approximately how fast does the peak-value $m_{peak}$ move as $k$ grows?  I briefly looked at the , but this involves distinct primes and I don't really know much number theory, so any help would be greatly appreciated! $f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$ if $\lim_{z\to\infty}f(z)=0$ and $f$ is analytic in $|z|\ge1$ Let $f$ be analytic in $|z|\ge1$ and  $\lim_{z\to\infty}f(z)=0$. Then is $f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$,$\forall |z|\ge2$, where $\Gamma$ is the circle of radius $2$ with centre at the origin.   I think yes, by the Cauchy integral formula. But, how do we ensure that the non analyticity of $f$ in $|z|&lt;1$ dosent affect the formula? Thanks beforehand. Workflow rule order of execution Let's stop displaying a user's accept rate How does the tmp folder work? It stores lots of junk files and takes space My tmp folder is set to /sites/default/files/tmp/ and I checked what's inside and for example there is folder inside called /update-cache-b2903cc4/ it keeps all modules in .zip that I installed via 'Install from a URL'. So why are those modules .zip files are kept there in tmp folder, isn't it a waste of storage? I don't really like it when the site is full of unnecessary files.  Am I supposed to delete these by myself all the time? I have this table but check constraint is throwing me an error:     Column CHECK constraint for column 'MedicamentRegulated' references another column, table 'Medicaments'.   I've seen this as allowed in SQL Server but I don't know why it doesn't work :/  CREATE TABLE Medicaments ( MedicamentID INT NOT NULL IDENTITY(1,1) PRIMARY KEY, Name VARCHAR(50), DateValidity DATE, MedicamentRegulated BIT CHECK((MedicamentRegulated = 1 AND DateValidity IS NOT NULL)    OR MedicamentRegulated = 0) ) How to express trigonometric equation in terms of of given trigonometric function? My Command is below and its print Free CPU and Used CPU below format.  I want to restrict awk output to 2 decimal place.  grep 'cpu ' /proc/stat | awk '{usage=($2+$4)*100/($2+$4+$5)} {free=($5)*100/($2+$3+$4+$5)} END {print "Used Cpu:",usage "%"} {print "Free Cpu:",free "%"}'   Output :   Free Cpu: 98.9588%  Used Cpu: 1.04125%   Desired output :  Free Cpu: 98.95%  Used Cpu: 1.04% How to both display a command line's output on console and save the output into a text file? Integral of $\sin(e^t)dt$ Let $f(x) = \int_x^{x+1} \sin(e^t)dt$. Show that $$ e^x|f(x)| &lt; 2 $$ and that $$ e^xf(x) = \cos(e^x) - e^{-1}\cos(e^{x+1}) + r(x) $$ where $|r(x)| &lt; Ce^{-x}$ for some constant $C$.  I can do the first part easily. Integration by parts gives us $$ f(x) = \frac{\cos e^x}{e^x} - \frac{\cos e^{x+1}}{e^{x+1}} - \int_{e^x}^{e^{x+1}} \frac{\cos u}{u^2}du, $$ so substituting $\cos u = 1$ and $\cos u = -1$ gives \begin{align*} \cos e^x - 1 + \frac{1}{e}\left(1- \cos e^{x+1}\right) &amp;\leq e^xf(x) \leq \cos e^x + 1 - \frac{1}{e}\left(\cos e^{x+1} + 1\right)\\ -2 &amp;&lt; e^xf(x) &lt; 2 \end{align*} So that $e^x|f(x)| &lt; 2$.  I get stuck on the second part, though. From integration by parts, I have $e^xf(x) = \cos e^x - e^{-1}\cos e^{x+1} - e^x\int_{e^x}^{e^{x+1}} \frac{\cos u}{u^2}du$, so it reduces to showing that $$ \left|\int_{e^x}^{e^{x+1}} \frac{\cos u}{u^2}du\right| &lt; Ce^{-2x} $$ for some constant $C$. However, I always end up with a constant for $Ce^{x}$ rather than for $C$. I'm completely new and lost in Ubuntu 12.04, I'm having trouble navigating to my Sublime folder which I know is in my /opt folder. Yet I've tried to no avail to navigate into it via Terminal window. The directory is colored in blue, no idea what this means...but I am able to access the folder using the GUI explorer...?  k@k-Ubuntu:~$ cd /opt k@k-Ubuntu:/opt$ ls Sublime Text 2 k@k-Ubuntu:/opt$ cd Sublime Text 2 bash: cd: Sublime: No such file or directory k@k-Ubuntu:/opt$ cd /Sublime Text 2 bash: cd: /Sublime: No such file or directory Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$ How to do this the fastest way? Given, $\{x_n\}$ and $\{y_n\}$ are two real valued sequences such that $\lim_{n\to\infty} (x_{n}^{3}-y_{n}^{3})=0 $ Then is it always true that $\lim_{n\to\infty} (x_n-y_n) =0 $ ? I tried by splitting $ (x_{n}^{3}-y_{n}^{3}) $, but that didn't help as $\lim_{n\to \infty} (x_{n}^{2}+x_{n}y_{n}+y_{n}^{2}) $ can be $0$ or non zero or doesn't exist. Test for existence of nested JavaScript object key Undocumented increase in minimum bounty value? I recently discovered that if I have posted an answer to a question, the minimum bounty I can offer for that question is +100 (instead of the usual +50).   Temporarily deleting the answer reinstates the +50 bounty (a workaround!).  As far as I can tell, this is not mentioned in  , , or . I've also tried  but didn't get very far.  So:   Should the FAQs be updated to include this information? Why have this rule in the first place if it's so easily bypassed?     Update:  This is indeed a new feature (introduced in ) to .This is mentioned explicitly in a few   but, as far as I can tell, not in FAQs. This information is therefore out of reach of mere mortals who do not have a finger on the pulse of MSO/SE.   The loophole mentioned above , but as of this update, can still be exploited (tested on superuser).  We can haz bugfix and faq update, pliz? Some Data Failed to Reconstruct, Motion Tracking I added tracks and stuff, but then when I tried to solve motion track, there was a error that said:     Some Data Failed to Reconstruct   and it also said to look into the console for details, which I did. This is what the console said:    By my understanding, my video is 372 frames long, and Blender is cutting the footage at 318 for some reason. Here is the image of the scene:    Why? Here are links to  and also . Also the next morning I did a camera solve, but then most of the frames went red. Diagnostics for logistic regression? How to compute moments of log normal distribution? The computed moments of log normal distribution can be found . How to compute them? How can I add a "C" wire to my thermostat?