Prove that the sum to n terms of the Sequence:  $1^2/(1×3),2^2/(3×5),3^2/(5×7),...$ is $  n(n+1)/2(2n+1).$  Im having trouble with this question, firstly ive begun by stating that p(n) denotes the statement.  Then to prove that p(1) holds where $n = 1$ $n(n+1)2(2n+1).$ = 1/3 and $1^2/1*3$ = $1/3$ therefore true  Now i know I have to show $n=k$ and therefore $n=k+1$ but how can i do this? I'm very tired so clear instruction would be great.
Prove that the sum to $n$ terms of a sequence:  $$\frac{1^2}{1\times 3}+\frac{2^2}{3\times 5}+\frac{3^2}{5\times 7}+\cdots $$ is  $$ \frac{n(n+1)}{2(2n+1)} $$
How can I combine hashes in Perl? What is the best way to combine both hashes into %hash1?  I always know that %hash2 and %hash1 always have unique keys.  I would also prefer a single line of code if possible.    $hash1{'1'} = 'red'; $hash1{'2'} = 'blue'; $hash2{'3'} = 'green'; $hash2{'4'} = 'yellow'; There are $2n+1$ people. For each $n$ people there is somebody who is friend with each of them. Prove there is a "know-them-all" person. I have a following problem.     There are $2n+1$ people in the room. For any group of $n$ people there is a person (different from them) who is friends with each person in this group. Prove that there is a person, who knows all $2n$ other people.   One can easily see, that $\min_v \deg v = n$. From this we can get a lower bound for number of edges $N_e$: $$ N_e = \frac{1}{2}\sum_v \deg v \geq \frac{(2n+1) \cdot n}{2} $$ I do not know where to move from this point (and do not think, that this is the right direction) and pretty stuck right now. Can you give a hint? Is there a robust command line tool for processing csv files? I work with CSV files and sometimes need to quickly check the contents of a row or column from the command line.  In many cases cut, head, tail, and friends will do the job; however, cut cannot easily deal with situations such as  "this, is the first entry", this is the second, 34.5   Here, the first comma is part of the first field, but cut -d, -f1 disagrees. Before I write a solution myself, I was wondering if anyone knew of a good tool that already exists for this job.  It would have to, at the very least, be able to handle the example above and return a column from a CSV formatted file.  Other desirable features include the ability to select columns based on the column names given in the first row, support for other quoting styles and support for tab-separated files.  If you don't know of such a tool but have suggestions regarding implementing such a program in Bash, Perl, or Python, or other common scripting languages, I wouldn't mind such suggestions. I am using different operating systems on different machines on different locations. I am using a USB flash disk to keep my files (source codes etc.).  I am new to Mac OS X and now I realize there are lots of files created with a dot underscore ._ prefix.  Questions  Why are ._ files created?   What are their purpose?   How can I avoid them? bash aliases equivalent for powershell? By default my Windows PowerShell starts in C:\Users\Santosh, my XAMPP installation is in D:\ so the htdocs folder is located at D:\xampp\htdocs. If I have to edit something in htdocs folder then I have to type full cd D:\xampp\htdocs\ (autocompletion is not so kind) then edit that file.  If this PowerShell were a Bash I would do this in .bash_aliases file:  alias htdocs='cd D:\xampp\htdocs'  Is it possible to maintain Bash aliases like file and alias any command in PowerShell? Modified pattern does not see the pattern color option Noncoalescing and bouncing droplets if you set a dish filled with water onto a loudspeaker or other vibrator and make vibration, When it oscillates, it is possible to hold small droplets on its surface for a long time. I mean, when you put a droplet of water on vibrating water surface, it doesn't coalesce with the water and bounce on the surface of water.( see this video  )  As we know, drop reaches its minimal potential when coalescing with water ( that's why drops coalesce)  what exactly cause this? why doesn't the drop coalesce with the water? So far, I understand that I have to look at each Jordan block. How do I prove that for a Jordan block with a value of x, $J_n(x)$, the Jordan decomposition of $(J_n(x))^2$ is $J_n(x^2)$ ? I have a feeling this will lead me to the solution. Thanks for your time. How to show that all points are inside of unit circle? There are $n$ points on the plane. Any $3$ of them are inside of a unit circle. How to show that all points are inside of unit circle?  It is needed to prove that if there is a unit circle for each $3$  points which contains them, there is also a unit circle which contains all $n$ points. Are there any programs that will let me dictate (speech-to-text) to any program? Or at least to a text editor and text fields in Firefox.  I’m not interested in anything that uses an online service, it has to be on-device. It may be a commercial product. Automatically fit the graphics in the remaining space of a Beamer slide I'm just trying out Arduino Uno for the first time with 2 blinking LEDs on a breadboard. All the tutorials on the Internet seem to use a resistor. I do know the function of resistors, but does it really matter here? These LEDs are working just fine without a resistor. Limits involving factorials $\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$ I am trying to calculate the following limit  $$\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$$ where $k$ can be any number between $0$ and $N$.  I thought of the following:  If I take the logarithm of the expression then I get:  $$\lim_{N\to\infty} \left(\log(N!)-\log((N-k)!)-k\log N\right)$$    Using the Stirling formula this can be approximated as:  $$\lim_{N\to\infty} \left(N\log(N)-(N-k)\log(N-k)-k\log N\right)$$  Now there are two cases: If $k$ is $N$, then the second term vanishes and the remaining terms cancel. If $k$ is smaller than $N$, then I can drop the $k$ inside the second logarithm and all the terms cancel. So the limit $$\lim_{N\to\infty} \log\left(\frac{N!}{(N-k)!N^{k}}\right)=0$$ Which means that: $$\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}=1$$ I don't know if this is mathematically rigorous. Would like some help. Thanks I wanted to tweet something on the lines of "Stack&nbsp;Overflow is awesome". When I do this kind of stuff, I usually use a username (for example: ) or a hashtag ()  I found , but it seems it is from someone unrelated with Stack&nbsp;Overflow.  Is there a Twitter account I can reference, or should I use a hash-tag? Should I just point to the site?  Thanks!  (Notice: I've seen lots of questions regarding Twitter and Stack&nbsp;Overflow, but those are mostly about integration/login with Twitter for the users accounts; I'm just asking for a single account for Stack&nbsp;Overflow; probably for announcements and PR, which is different) How can I setup a Material in Cycles for Z-Transparency? How to set AUTO-SCROLLING of JTextArea in Java GUI? What are some of the rule changes that Pathfinder brings in comparison to D&amp;D 3.5?  I am curious what changes were made so I can explain to my group what it has to offer. Proving that a Hölder space is a Banach space I am trying to show that the Hölder space $C^{k,\gamma}(\bar{U})$ is a Banach space. To do this, I successfully proved that the mapping $\| \quad \| : C^{k,\gamma}(\bar{U}) \to [0,\infty)$ is a norm, by proving its properties.   But how do I show that the sequence $\{u_k\}_{k=1}^\infty \subset C^{k,\gamma}(\bar{U})$ converges to $u \in C^{k,\gamma}(\bar{U})$, that is, how do I show that $$\lim_{k \to \infty}\|u_k-u\|=0,$$ which would mean the normed linear space is complete, and hence a Banach space?  Here are the following taken from PDE Evans, 2nd edition, page 255:     Definition. The Hölder space $$C^{k,\gamma}(\bar{U})$$ consists of all functions $u \in C^k(\bar{U})$ for which the norm $$\|u\|_{C^{k,y}(\bar{U})}:= \sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})}$$ is finite.   Also from page 254,     Definitions. (i) If $u : U \to \mathbb{R}$ is bounded and continuous, we write $$\|u\|_{C(\bar{U})}:=\sup_{x\in U}|u(x)|.$$      (ii) The $\gamma$th-Hölder seminorm of $u : U \to \mathbb{R}$ is $$[u]_{C^{0,\gamma}(\bar{U})}:=\sup_{\substack{x,y\in U \\ x \neq y}} \left\{\frac{|u(x)-u(y)|}{|x-y|^\gamma} \right\},$$ and the $\gamma$th-Hölder norm is $$\|u\|_{C^{0,\gamma}(\bar{U})}:=\|u\|_{C(\bar{U})}+[u]_{C^{0,\gamma}(\bar{U})}.$$   This is all I got so far: \begin{align} \|u_k-u\|_{C^{k,\gamma}(\bar{U})}&amp;=\sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})} \\ &amp;= \sum_{|\alpha|\le k} \sup_{x\in U} |u_k(x)-u(x)|+ \sum_{|\alpha|=k} \sup_{\substack{x,y\in U \\ x \neq y}} \left\{\frac{|[u_k(x)-u(x)]-[u_k(y)-u(y)]|}{|x-y|^\gamma} \right\}. \end{align} Now, where can I go from here, to show that $\lim_{k \to \infty}\|u_k-u\|_{C^{k,\gamma}(\bar{U})}=0$? The sequence is Cauchy, and I have to use that fact somehow. If the dot product between two vectors is $0$, are the two linearly independent? If we have vectors $V$ and $W$ in $\mathbb{R^n}$ and their dot product is $0$, are the two vectors linearly independent?  I can expand $V_1 \cdot V_2 = 0 \Rightarrow v_1w_1+...+v_nw_n = 0$, but I don't understand how this relates to linear independence. In the simple linear regression case $y=\beta_0+\beta_1x$, you can derive the least square estimator $\hat\beta_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}$ such that you don't have to know $\hat\beta_0$ to estimate $\hat\beta_1$  Suppose I have $y=\beta_1x_1+\beta_2x_2$, how do I derive $\hat\beta_1$ without estimating $\hat\beta_2$? or is this not possible?