Note: I am asking this question as a simple introductory question to proofs by induction, to which I will give also my formal answer (which should be correct, if not, please comment) for future visitors of this website.  I know that to prove something by induction, we need formally 3 steps (2, if we considere inductive hypothesis and inductive step together):   Base case or basis:  We determine what is the base case for our statement $P(n)$, and we immediately prove it. Inductive hypothesis:  We assume our statement $P(n)$ is true for $n$. Inductive step:  We try to prove that our statement is also true for $P(n + 1)$. If we can prove that, we can also prove that our statement is true for $n + 2$ (we don't need to do it), and so on.  This step is usually the most difficult part, because it involves some algebraic manipulation, or some imagination in some way. For this last reason, it might be helpful watching other similar proofs by induction to solve the current one.   For example, suppose our $P(n)$ is the following:  $$\frac{n(n + 1)(2n + 1)}{6} = 0^2 + 1^2 + 2^2 + 3^2 + ... + n^2$$  which is basically saying that the sum of all squares of the natural numbers from $0$ to $n$ can be determined by the formula on the left side of the equation.  How would we prove $P(n)$ is true for all $n \in \mathbb{N}$?  For more information about the proof by induction, see the Wikipedia page about induction called .
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:  $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$  I really have no idea why this statement is true. Can someone please explain why this is true and if possible show how to arrive at one given the other?
C# Speech Recognition - Is this what the user said? I have need to write an application which uses a speech recognition engine -- either the built in vista one, or a third party one -- that can display a word or phrase, and recognise when the user reads it (or an approximation of it).  I also need to be able to switch quickly between languages, without changing the language of the operating system.  The users will be using the system for very short periods.  The application needs to work without the requirement of first training the recognition engine to the users' voices.  It would also be fantastic if this could work on Windows XP or lesser versions of Windows Vista.  Optionally, the system needs to be able to read information on the screen back to the user, in the user's selected language.  I can work around this specification using pre-recorded voice-overs, but the preferred method would be to use a text-to-speech engine.  Can anyone recommend something for me? Is there any way to specify a suggested filename when using data: URI? If for example you follow the link:  data:application/octet-stream;base64,SGVsbG8=  The browser will prompt you to download a file consisting of the data held as base64 in the hyperlink itself.  Is there any way of suggesting a default name in the markup?  If not, is there a JavaScript solution? Use Complete Induction of set theory to prove $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}} > \sqrt{n}$. Function that returns the second argument of Reap Let $A [a_{i,j}]$  and $B =[b_{ij}]$   be two diagonal matrices in   $M_n(F)$. Let $i \ne j \in \left\{1, \ldots, n \right\}$. Using the   product rule, we obtain      $$(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$      We claim that $a_{ik}b_{kj} =0$ for all $k \in \left\{1, \ldots, n \right\}$ thus $(AB)_{ij}=0$ for all $i \ne j$ and $AB$ is diagonal. To    prove the claim, note that since $i \ne j$ , we have $i \ne k$ or $    k\ne j$. Thus either $a_{ik} = 0$ (since $A$ is diagonal) or $b_{kj} =  0$ (since $B$ is diagonal), thus in all cases $a_{ik}b_{kj}=0$ and the   claim is proved.   Where does the bit "we have $i \ne k$ or $k\ne j$" come from? I understand that since $i \ne j$, and $k$ and $j$ run over the same set of natural numbers, we also have $i \ne k$. But the "or $k \ne j$" baffles me. I understand that $k \ne j$ implies that we can have $i=k$. But that's from the other direction ($\Longleftarrow).$  Also, when it says $a_{ik}=0$ or $b_{kj}=0$ it's talking about when $i \ne k$ or $k \ne j$ right? Many thanks. Discontinuous derivative. Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval. I came across this :     Unbelievable as it may seem, many individuals that fought in the   American Revolution were still alive in 1839, the year the world was introduced to photography. (choose the error - the 'answer' is in bold)   It noted that the reason "that fought" is wrong is that 'The relative pronoun “that” may not be used to refer to people (“individuals”) and should be replaced with the relative pronoun “who.”'   I had thought that the word "that" could refer to both people and objects (while which was reserved for objects and who reserved for people). Am I wrong? Is there any UNIX variant on which a child process dies with its parent? My understanding is that when using a Bayesian approach to estimate parameter values:   The posterior distribution is the combination of the prior distribution and the likelihood distribution. We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution). Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean.   But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it? How come everyone easily takes or say the name of "Voldemort" without any fear in Scorpius's world? Give a sequence such that root test works while ratio test fails We have an instance of SQL Server 2008 installed on a customer server. The customer's IT department is in charge of the various backups of the machine.  The SQL Server log shows there is a full backup every day at 7 PM but we can't find evidence of any plan scheduled in SQL Server.  The technicians we contacted couldn't tell us if there was some sort of automatic backup, all they told is that the entire machine is being backup up.  Using a script I found in  I found out that the physical device name is a GUID and that this means that this is an external backup process:  USE [msdb] GO  SELECT      [bs].[database_name],      [bs].[backup_start_date],      [bs].[backup_finish_date],      [bs].Server_name,     [bs].user_name AS [BackupCreator] ,     [bmf].physical_device_name FROM msdb..backupset bs   INNER JOIN msdb..backupmediafamily bmf ON [bs].[media_set_id] = [bmf].[media_set_id]  ORDER BY [bs].[backup_start_date] DESC   This is a row from the above query:     db_name 2015-09-16 19:01:23.000   2015-09-16 19:01:28.000SERVER_NAME  NT AUTHORITY\SYSTEM{424F084A-F35D-4A66-8FC7-072268A89A77} 5   Moreover the backup start and finish date spans only for 5 seconds, so I guess it's clear it's not a job of sql server.  This is a line from the log:     2015-09-03 19:02:30.71 Backup   Database backed up.   Database: db_name, creation date(time): 2012/10/12(20:52:11), pages dumped: 3290021,   first LSN: 276028:152755:172, last LSN: 276028:152827:1, number of dump devices: 1,   device information:   (FILE=1, TYPE=VIRTUAL_DEVICE: {'{95380B0A-D50B-408F-B95F-1AB8975BA7F8}5'}).   This is an informational message only. No user action is required.   So, since they can't help us what can I do to track down the process responsible of the backups? We need this because we want to coordinate the backup in an ordered manner, and do the transaction log backups too in order to keep the log at a reasonable size (now we do shrink the log every week, and this is not the good way to go). Non-uniform distribution of p-values when simulating binomial tests under the null hypothesis I heard that under the null hypothesis the p-value distribution should be uniform. However, simulations of binomial test in MATLAB return very different-from-uniform distributions with mean larger than 0.5 (0.518 in this case):   coin = [0 1]; success_vec = nan(20000,1);  for i = 1:20000     success = 0;     for j = 1:200         success = success + coin(randperm(2,1));     end     success_vec(i) = success; end      p_vec = binocdf(success_vec,200,0.5);     hist(p_vec);   Trying to change the way in which I generate random numbers didn't help.  I would really appreciate any explanation here. A hard problem on exponential integration Suppose $a : [0 , 1] \to \Bbb R$ is an infinitely smooth function. For $\lambda\ge1$, define $$F(\lambda) := \lambda \int_0^1 e^{\lambda t} a(t) \, dt.$$ If $\sup_{\lambda\ge1}|F(\lambda)|\lt\infty$, then $a$ is the identically zero function. Below are the some results I have derived:   Derivatives of any order of $a$ vanishes at $t = 1.$ For any $\delta\lt1$, $a(t)$ has a zero in the open interval $(\delta , 1).$  The same is true for all the derivatives of $a(t)$. This tells us that the $n^\text{th}$ derivative of $a$ has infinitely many distinct zeros for all natural $n$. If $a$ is analytic, then I can show that $a(t)\equiv 0.$ If $a(t)\ge0$ on $[0 , 1]$, then it is obvious that $a\equiv 0.$   I would appreciate any hint.  Actually $(1)$ and $(3)$ follows from $(2)$. For $(2)$, suppose that $a(1)\gt0$ (the case that $a(1)\lt0$ can be argued in the same way as the following), then there exists $\delta\lt1$ such that $a(t)$ is strictly positive on $I = [1-\delta , 1]$. Then write $F(\lambda) = \lambda\int_0^{1-\delta}e^{\lambda t}a(t)dt + \lambda\int_{1-\delta}^1e^{\lambda t}a(t)dt$. Now there exists $c\gt0$ such that $a(t)&gt;c$ on $I$ since $I$ is compact, so that the second term is bounded below by $c (e^{\lambda}-e^{\lambda(1-\delta)})$. But the first term is only $O(e^{\lambda(1-\delta)})$, so this contradicts the fact that $F(\lambda)$ is bounded. Morever, using the same idea and integration by parts, one can see that $n^{th}$ derivative of $a$ must vanish at $1$. Understanding etoolbox \ifdefempty and \ifdefvoid I am trying to understand the  \ifdefempty and \ifdefvoid macros. The documentation says that \ifdefempty     Expands to true if the control sequence is defined and is a   parameterless macro whose replacement text is empty, and to false   otherwise.   Similarly \ifdefvoid     Expands to true if the control sequence is undefined, if is a macro   whose meaning is \relax, or a parameterless macro whose replacement   text is empty, and to false otherwise.   From these descriptions I thought that \def\myempty{}, \def\myrelax{\relax}, and \def\myemptyempty{\myempty} would all be both "empty" and "void", but \def\myempty{} is the only one that is either "empty" or "void". What am I missing?  \documentclass{minimal} \usepackage{etoolbox}  \begin{document}  \def\myempty{} \def\myrelax{\relax} \def\myemptyempty{\myempty}   \ifdefempty{\myempty}{Empty}{Not empty}  \ifdefempty{\myrelax}{Empty}{Not empty}  \ifdefempty{\myemptyempty}{Empty}{Not empty}  \ifdefvoid{\myempty}{Void}{Not void}  \ifdefvoid{\myrelax}{Void}{Not void}  \ifdefvoid{\myemptyempty}{Void}{Not void}  \end{document} How can I revert a config file back to the originally installed version after I have edited it? I want to revert a file in /etc/ back to its originally-installed form. I have edited the file since it was installed. How can I revert this config file? Apt is smart enough not to overwrite edited config files, so how do I tell it that I want it to do so?  For argument's sake, let's say that I want to revert the file /etc/foo.conf from the package foo. How should tiny $p$-values be reported? (and why does R put a minimum on 2.22e-16?) For some tests in R, there is a lower limit on the  calculations of $2.22 \cdot 10^{-16}$. I'm not sure why it's this number, if there is a good reason for it or if it's just arbitrary. A lot of other stats packages just go to 0.0001, so this is a much higher level of precision. But I haven't seen too many papers reporting $p &lt; 2.22\cdot 10^{-16}$ or $p = 2.22\cdot 10^{-16}$.   Is it a common/best practice to report this computed value or is it more typical to report something else (like p &lt; 0.000000000000001)? How do I print bold text in Python? How to repeat a string a variable number of times in C++? I want to insert 'n' spaces (or any string) at the beginning of a string in C++. Is there any direct way to do this using either std::strings or char* strings?  E.g. in Python you could simply do  &gt;&gt;&gt; "." * 5 + "lolcat" '.....lolcat' I would like to learn how to use it so I can start to use the terminal more.