Corollary C is derived from $E_(x) = \frac{f''(x)}{2} \cdot (x-a)^2$  I'm having serious issues understanding this problem and I'm just not getting it right. I'm preparing for a test so this isn't really 'homework' so you can give as hard hints as you want. Thanks!
Question: Can anyone shed some light on this formula? I can't find any information on it.  It has three corollaries that I also need to understand:
So I have learned in class that light can get red-shifted as it travels through space. As I understand it, space itself expands and stretches out the wavelength of the light. This results in the light having a lower frequency which equates to lowering its energy.  My question is, where does the energy of the light go? Energy must go somewhere!   Does the energy the light had before go into the mechanism that's expanding the space? I'm imagining that light is being stretched out when its being red-shifted. So would this mean that the energy is still there and that it is just spread out over more space? Installing Reaver 1.4 I am trying to install reaver 1.4, but when I run ./configure ; make I get this message at the end  int ret_val = EXIT_FAILURE, r = 0;   I have searched a fair bit and cannot find an answer, any help would be appreciated. I apologize if this has been answered and I've just failed to find it. How to extract Time from Date/Time field within formula How do I get started in Astrophotography? $f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable Product of Non-Disjoint Cycles I am trying to learn how to find the product of non-disjoint cycles, as you may have guessed from the title. I have the basic idea, but I do not understand it entirely.  I am trying to find $(1352)(256)$ and $(1634)(1352)$, multiplying from right to left.  Please give me some hints/suggestions.  Thank you. How to install the Pantheon desktop environment? This is a  about RAID levels.    What are:   the RAID levels typically used (including the RAID-Z family)?  deployments are they commonly found in? benefits and pitfalls of each? Where do I use delegates? What are some real world places that call for delegates?  I'm curious what situations or patterns are present where this method is the best solution.  No code required. I quite enjoy paying attention to how I design my documents and presentations. I usually spend hours thinking over and designing my slides for a workshop or presentation, so that they are aesthetically pleasing and as intuitive as possible. Likewise I recently started revising my CV I figured and I wanted to make it stand out a bit more. (Just to make it clear I don't mean making a clown of a document but just better use of colors, contrast and design elements.)  I have long wondered whether or not this is something that can backfire, since most documents in academic context are extremely plain, at least in my experience. It's very common to see the default Powerpoint slides (white bg, black arial text) or .   My question is as follows: is putting time and effort into design of academic documents something that can backfire? Will I risk being prejudged with first impressions such as "well he put much effort in the presentation his documents, perhaps because the content is sub-par"?     I realize that the question might be somewhat subjective from person to person but I encourage everyone to consider it in terms of this SE blog entry: . Here is the problem: From any IP address not belonging to your mail server:  telnet me.myemailserver.com 25    helo me.someserver.com mail from: &lt;yourusername@mydomain.com&gt; rcpt to: &lt;yourusername@mydomain.com&gt; data This is spam.  Buy my stuff. .   I'm using Postfix.  I'm having a problem finding a solution to requiring SMTP-AUTH for email claiming to be from mydomain.com.  Googling around, this guy has identified the same problem (where I cut-n-paste with some modifications) the above example from:   This link   gets close to a solution but it has a side effect of requiring SMTP-AUTH for mail not from mydomain.com.  For mail not claiming to be from mydomain.com, I would do the usual RBL and Spam filtering.  In short, I want to reject mail to local domains (mydomain.com) from outside/unauthenticated clients claiming to be from local domains (mydomain.com).  This is what I tried: I've tried both permit and reject as the default.  Here is exact excerpt from my main.cf:  smtpd_recipient_restrictions = reject_unauth_pipelining,                            permit_sasl_authenticated,                            check_recipient_access pgsql:/etc/postfix/pgsql-recipient.cf,                            reject_unauthenticated_sender_login_mismatch,                            reject_unauth_destination,                            reject_unlisted_recipient,                            check_sender_access pgsql:/etc/postfix/pgsql-sender.cf,                            reject_unlisted_sender,                            reject_invalid_hostname,                            reject_non_fqdn_hostname,                            reject_non_fqdn_sender,                            reject_non_fqdn_recipient,                            reject_unknown_sender_domain,                            reject_unknown_recipient_domain,                            reject_rbl_client cbl.abuseat.org,                            reject_rbl_client sbl.spamhaus.org,                            reject_rbl_client sbl-xbl.spamhaus.org,                            reject_rbl_client bl.spamcop.net,                            reject_rbl_client dnsbl.njabl.org,                            reject_rbl_client blackholes.wirehub.net,                            reject_rbl_client relays.mail-abuse.org,                            reject_rbl_client dialups.mail-abuse.org,                            reject_rbl_client blackholes.mail-abuse.org,                            reject_rhsbl_sender dsn.rfc-ignorant.org,                            (reject and permit both tried here) How to set custom keyboard shortcuts from terminal? If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$ Here's the problem in full. I've stared at it for hours and can't get anywhere, so a hint would be nice.  Suppose that $f:\mathbb N\rightarrow\mathbb R$. If $$\lim_{n\rightarrow\infty}f(n+1)-f(n) = L$$ prove that $\lim_{n\rightarrow\infty}f(n)/n$ exists and equals $L$.  This problem is marked with the word "Cauchy", so I'm guessing Cauchy sequences or something related to them will be necessary, but I can't figure out what.  Edit: It appears as if the Stolz-Cesaro Theorem makes this quite easy. Unfortunately though, we have not discussed this, so I won't be able to use this. Center Gravity in Unity Translation help review queue? Story about lost colonized planet sent into orbit around "free" star how I can draw an skew cylinder (rotate around x axis) and fill it (in 3D tikz)? Problem applying L'Hopitals rule. Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$  Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$.  I would however like to learn the technique of dealing with surds such as these, especially in trignometric problems as I have a lot of similar problems and I don't have a clue as to how to deal with those.  Hints please!  EDIT:  What I have done using half angles is this: (and please note, for convenience, I am dropping the degree symbols. The angles here are in degrees however).  I know that $$ \cos 15 = \dfrac{\sqrt3+1}{2\sqrt2}$$  So,   $$\sin7.5 = \sqrt{\dfrac{1-\cos 15} {2}}$$ $$\cos7.5 = \sqrt{\dfrac{1+\cos 15} {2}} $$  $$\implies \cot 7.5 = \sqrt{\dfrac{2\sqrt2 + \sqrt3 + 1} {2\sqrt2 - \sqrt3 + 1}} $$ Number of non-decreasing functions? Let $A = \{1,2,3,\dots,10\}$ and $B = \{1,2,3,\dots,20\}$. Find the number of non-decreasing functions from $A$ to $B$.  What  I tried: Number of non-decreasing functions = (Total functions) - (Number of decreasing functions) Total functions are $20^{10}$. And I think there are ${20 \choose 10}$ decreasing functions. Since you choose any $10$ codomain numbers and there's just one way for them to be arranged so that the resultant is a decreasing function. However my answer doesn't match. Where am I going wrong? How can I directly compute the non-decreasing functions like without subtracting from total?