let $x&gt;0$ And $f(x) =\dfrac{\sin x}{x}$ prove that for every $n$ :  $$|f^{(n)}(x)|&lt;\frac{1}{n+1}$$    for $f^{(1)}$ we have :  $$\frac{x\cos x-\sin x}{x^2}&lt;\frac{1}{2}$$    for $f^{(2)}$ we have :  $$\frac{x^2(-x\sin x)-2x(x\cos x-\sin x)}{x^2}&lt;\frac{1}{3}$$  Now what ?
Suppose $x&gt;0$ and we have function $f(x)=\frac{\sin x}{x}$ how can we show $$\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$$ I need a hint to show this property .Thanks in advance . I tried for $n=1 ,2$  by finding maximum of $|f'| ,|f''|$ but I get stuck to show for $n$
Why is sum of squared residuals non-increasing when adding explanatory variable? In my econometric textbook(Introductory Econometrics) covering OLS, the author write, "SSR must fall when another explanatory variable is added."  Why is it? Help with this trigonometry problem Prove the given identity.  $$\left(\sin\frac {9\pi}{70}+ \sin\frac {29\pi}{70} - \sin\frac {31\pi}{70}\right) \left(\sin\frac {\pi}{70}-\sin\frac {11\pi}{70} - \sin\frac {19\pi}{70}\right) =\frac {\sqrt {5} -4}{4}$$  Please help, I could not gather enough ideas, even on how to get to the first step. I thought of using the transformation formula (from sum to product) but did not find a fruitful result.   UPDATE 1.:  $\left(-\frac{1 +\sqrt{5}}{8} + \frac{1}{8} \sqrt{14\left(5 - \sqrt{5} \right)}\right)\left(-\frac{ 1+ \sqrt{5}}{8} -\frac{1}{8} \sqrt{14 \left(5 -\sqrt{5} \right)}\right) $,    I noticed that Alexis had got to this point.  Can anyone explain me how did he get here, with calculations. Please help.  Thanks in Advance. TikZ: Having circle borders only grown to the outside Is it feasible to transform each variable differently while doing multiple regression Apollonius circle, its radius and center Probability Math Question...? Bag A contains 3 white and 2 red balls. Bag B contains 6 white and 3 red balls. One of the two bags will be chosen at random, and then two balls will be drawn from that bag at random without replacement. What is the probability that the two balls drawn will the be the same color? Express your answer as a common fraction. Why doesn't 'find' show this file? Use column alias in GROUP BY I have a query where I chain two columns. Why does the alias not work in the GROUP BY clause but in ORDER BY clause, it does work? How can I write the select correctly?  SELECT   KOS_VER_ID        AS "Vertrag"   , WHR_ISO_3_CODE  AS "Waehrung"   , KOS_KOA_ST_KZN || '  -  ' || ST_LANGBEZ_EN as "Kostenart"   , SUM (KOS_BETRAG) AS "Summe pro KOA" FROM   KOSTEN   , WAEHRUNG   , SCHLUESSELTABELLE WHERE    KOSTEN.KOS_VERHI_WHR_ID = WAEHRUNG.WHR_ISO_ID   AND KOSTEN.KOS_KOA_ST_KZN = SCHLUESSELTABELLE.ST_ID   AND KOS_VER_ID in (2509, 2510, 2511) GROUP BY   KOS_VER_ID   , WHR_ISO_3_CODE   , KOS_KOA_ST_KZN || '  -  ' || ST_LANGBEZ_EN ORDER BY   "Vertrag"   , "Kostenart" ; Renaming accessor/mutator methods in Eclipse? Is there any way to automatically rename accessor/mutator when a variable they get/set gets refactored -> renamed (Eclipse 3.4)? It's a shame to type out an entire question, press post and then:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Could you put some sort of notice on the page itself before submitting it to let you know the question won't be going through?  Or perhaps tell me how many questions I've asked in the last 30 days if it's >0?  Edit  This is not a complaint or statement about the limit, just a request that the error handling be done in a slightly better way to save people a lot of time. What have to be normally distributed: groups or whole sample? It is a general and maybe basic question, but it will help me to avoid a mistake. To test the dependence of continuous variable on nominal with parametric test (t-test, ANOVA), "the data has to be normally distributed" (distribution have to be not much violated from normal).   But I don't clearly understand: I have to test the normality of the continuous data together (whole sample) or the normality of data in every single group?   And if my groups have small sizes (n=4-6), and therefore I fail to reject null, does this indicate a permission to use parametric methods?  Can anyone provide me a link to some good examples of statistical (biomedical) hypothesis testing with R (google did not helped)? Can only search social network messages in the Dash after 13.10 upgrade If I live in the US, for example, where current interest rates are practically nil, how can I take advantage of the fact that ? Was Yoda wrong about the limits of telekinesis? In Empire Strikes Back, Yoda  that an object's size doesn't matter and that the difficulty of levitating an X-Wing was only in his mind. If the upper limits of telekinesis are determined by belief, what's to stop a Jedi/Sith from pulling a moon out of orbit, moving an entire solar system, or splitting an atom with the Force? Hide the term id on autocomplete widget How do you prevent web pages from appearing in the Google search index? Why are the even and odd Regge trajectories degenerate? I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan Why does observation collapse the wave function? In one of the first lectures on QM we are always taught about Young's experiment and how particles behave either as waves or as particles depending on whether or not they are being observed. I want to know what about observation causes this change? Show that a finite cyclic group of order $n$ has exactly one subgroup of order $d$ dividing $n$ and these are all the subgroups it has My attempt  Note- Addition is under $mod$ $d$  Let $G = (a^0,a^1,a^2.....a^{n-1})$  $n=kd$  Let $H_d = (a^{0k},a^{1k},a^{2k}......a^{{(d-1)}k})$  For $a^{z_1k},a^{z_2k} \in H_d$  $a^{z_1k}a^{z_2k} = a^{{(z_1+z_2)}k} \in H_d$  $H_d$ is closed  For $a^{zk} \in H_d$  $a^{zk} (a^{zk})^{-1}=a^0$  $(a^{zk})^{-1} = (a^{-1})^{zk}$  Since $a^{-1}$ is $a^{n-1}$ we can rewrite it as   $(a^{zk})^{-1} = (a^{n-1})^{zk}=a^{z(n-1)k} \in H_d$  Since $(a^{zk})^{-1} \in H_d$ and $H_d$ is closed, $H_d$ is a subgroup.  I can easily prove the 3rd part by using Lagrange's theorem.  I have no idea how to prove that only $1$ subgroup of order $d$ exists. Maybe by proving that the elements of the $2$ groups have to be the same?  I would really appreciate if someone could verify my proof so far and give some hints to complete the proof.