how to store photos and videos in Core Data?
Core Data - Storing Images (iPhone)
Probability of winning the game "1-2-3" I am trying to prove the following by Mathematical Induction:  $$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$   My proof by Induction follows: First prove $P(1)$ is true,  $$\sum_{i=1}^1 i\cdot i! = (1+1)! - 1$$  Then, for all integers $k &gt;= 1$, if $P(k)$ is true then $P(k+1)$ is also true,  $$ \sum_{i=1}^k i\cdot i! = (k+1)! - 1$$ Then, we must show that $P(k+1)$ is true   $$ \sum_{i=1}^{k+1} i\cdot i! = ((k+1)+1)! - 1\\ \sum_{i=1}^{k+1} i\cdot i! = (k+2)! - 1$$  I am currently struggling with the next step to show that $P(k)$ is equal to $P(k+1)$. How would I finish this prove by induction? A new search engine for Stack Exchange Why is SQL Server consuming more server memory? Prove $\lim_{x\to0}\sqrt{4-x}=2$ using the precise definition of limits. (Epsilon-Delta) Prove $$\lim_{x\to0}\sqrt{4-x}=2$$ using the precise definition of limits. (Epsilon-Delta)  I am not sure how to link $0&lt;\left |x  \right |&lt;\delta$ with $\left |\sqrt{4-x}-2\right |&lt;\epsilon$ .  EDIT (Trying it out now)  I worked till here, then I basically got stuck. How can I make something that looks like this?  --------   Text   ---------  where the horizontal rules extend to the margin on either side? Calculating $\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}$ I want to calculate the sum:$$\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}=$$  $$\sum_{n=1}^\infty\frac{n}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{n+1-1}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{n+1}{(n+1)!}-\sum_{n=1}^\infty\frac{1}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{1}{n!}-\sum_{n=1}^\infty\frac{1}{(n+1)!}.$$  I know that $$\sum_{n=0}^\infty\frac{1}{n!}=e$$  so $$\sum_{n=1}^\infty\frac{1}{n!}=\sum_{n=0}^\infty\frac{1}{n!}-1=e-1$$  But, what can I do for $$\sum_{n=1}^\infty\frac{1}{(n+1)!}$$ ?  Am I allowed to start a sum for $n=-1$ ? How can I bring to a something similar to $$\sum_{n=0}^\infty\frac{1}{n!}$$? Is "Five dollars is all I have" correct? I want to emphasize that I only have 5 dollars, so can I use the sentence:     Five dollars is all I have.   Or must it be:      Five dollars are all I have.   Please give the explanation as well. I recently read a lot about neural networks using GRU or LSTM units. There are many easy to use frameworks like tensorflow that do not even require high knowledge about programming. Unfortunately, I never really found good information on how the training of those networks work. Simple backpropagation might probably not work for gated recurrent networks or is just too inefficient for networks with such a high number of variables to learn.  So my question is:  What are the state-of-the-art algorithms used for the initialization and training of neural networks with GRU or LSTM units? I am not looking for frameworks to use, but for initialization and update equations for the internal parameters. How could it possible to factorise $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$? How could it possible to factorize $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$?  I'm having a hard time starting the problem. Could anyone help me at this point? When I cut a layer using marquee tool, I get blur edges to cut side. How do I get sharp edges after cut? I am using rectangular marquee tool to cut a portion of the layer in photoshop cs5, but after cutting, the edges are not sharp, I get blur edges.      how can I get sharp edges? Regex to get string between curly braces Extending the User model with custom fields in Django What's the best way to extend the User model (bundled with Django's authentication app) with custom fields? I would also possibly like to use the email as the username (for authentication purposes).  I've already seen a   to do it, but can't decide on which one is the best. Showcase of beautiful typography done in TeX & friends If you were asked to show examples of beautifully typeset documents in TeX &amp; friends, what would you suggest? Preferably documents available online (I'm aware I could go to a bookstore and find many such documents called 'books'). Extra bonus for documents whose LaTeX source is available.  This is not an idle question. Seeing great examples of any craft is both educational and inspiring, let alone explaining why we prefer TeX to Word or other text editors.  For instance, I like how Philipp Lehman's  looks. I don't know enough LaTeX to realize how much customization was done, but the ToC looks polished.  Your nominations, please ... I've got a Wordpress site on our home intranet that has run into trouble now that the IP address has changed - The index page loads, but not the CSS and I can't log in to the site administration panel.  Unfortunately I am a bit behind on backups. Is there a way to get Wordpress to refer to the new IP address? How do I retrieve only the needed line from terminal output? Convolution is uniformly continuous and bounded Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function.  The definition of the convolution is $(f\ast K)(x)=\int_\mathbb{R}f(x-y)K(y)dy$.   There is the inequality $\|f\ast K\|_\infty\leq\|f\|_\infty\|K\|_1$, which yields that $f\ast K$ is bounded. But what about uniform continuity? I have an Ubuntu server and I've installed MySQL.  How would I use my two physical static IP addresses in my.cnf bind-address -- can I add two IP addresses? I'm trying to write a page that calls PHP that's stored in a MySQL database. The page that is stored in the MySQL database contains PHP (and HTML) code which I want to run on page load.  How could I go about doing this? Why does this CSS margin-top style not work?