Use the defintion of a logarithmic function to prove $$ \frac{1}{x+1} \le \ln(1+\frac{1}{x}) \le \frac{1}{x}$$ for any positive integer x.   ( Using defintion : $e^y=\ln^{-1}y$ )     Attempt:    Using the definition,  $e^y=\ln^{-1}y$ ,    I tried to prove the right hand side of the equality first by $ 1 +\frac{1}{x} \gt \frac{1}{x}$,     $\ln(1 +\frac{1}{x})\lt \frac{1}{x} \Rightarrow   e^{\ln(\frac{x+1}{x})} \le e^{\frac{1}{x}}$ but I am really stuck and lost on how I should do it so that one of the sides there is ln and the other side is the value and also for the equality to be $\le$ as $ 1 +\frac{1}{x} \gt \frac{1}{x}$  is strictly less than and is not equal.
I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this.   $$\frac 1{x+1}&lt;\ln\left(\frac{x+1}x\right)&lt;0.5\left(\frac 1x+\frac 1{x+1}\right)$$
How to randomly subset X% of selected points? What methods are available in ArcGIS 10.2 to randomly subset a selection of points.  For example, in the attached screenshot I am interested in keeping 20% of the selected points and deleting the rest. How do I change the computer name? C# 3.0 Auto-Properties - Is it possible to add custom behaviour? I would like to know if there is any way to add custom behaviour to the auto property get/set methods.  An obvious case I can think of is wanting every set property method to call on any PropertyChanged event handlers as part of a System.ComponentModel.INotifyPropertyChanged implementation.  This would allow a class to have numerous properties that can be observed, where each property is defined using auto property syntax.  Basically I'm wondering if there is anything similar to either a get/set template or post get/set hook with class scope.  (I know the same end functionality can easily be achieved in slightly more verbose ways - I just hate duplication of a pattern) Radius of convergence of $\sum_{n=0}^\infty n!x^{n^2}$ The question is to calculate the radius of convergence of $\sum\limits_{n=0}^\infty n!x^{n^2}$.  I dont know how to calculate R.O.C for such type of questions involving $n^2$ as power of x. How do we do this? Any help appreciated. I would like to take the full citation of a reference in my BibTeX file and write commentary in-line after it.  A \fullcitation{id} command would be perfect.  I would like to use it as:  \fullcitation{id1}   This paper talks about A with applications to B.  \fullcitation{id2}   This paper considers how C affects A.  Whenever the \fullcitation{id} command appears, I want it to be replaced with the full reference, including authors, title, journal, and date.  Is this possible? Primary ideals in Noetherian rings For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary or $(0)$ is a primary submodule.  Now for an ideal $\mathfrak{q} \subset R$ this definition gives that $\mathfrak{q}$ is $\mathfrak{p}$-primary iff $\text{Ass}(R/\mathfrak{q}) = \{\mathfrak{p}\}$. It can be shown that this is equivalent with $\mathfrak{p}^n \subset \mathfrak{q}$ for some $n\in \mathbb{N}$ and the condition \begin{align*} \forall r,s \in R: rs \in \mathfrak{q}, r\not\in \mathfrak{p}\Rightarrow s \in \mathfrak{q}. \end{align*}     Now I suspect that if $R$ is Noetherian, and if $\mathfrak{q}$ is $\mathfrak{p}$-primary, then we have $\sqrt{\mathfrak{q}} = \mathfrak{p}$. I have shown this for PID's, but does this generaly hold?    And if so, is every ideal $\mathfrak{q}\subset R$ then $\sqrt{\mathfrak{q}}$-primary?   Many thanks. How to convert light activated switch Can anyone teach me how to convert a light activated switch into  dark activated switch vice versa. Why quantum mechanics? Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before.  How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?  If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious.  Why should this be the case, or at least, why might we expect this to be the case?   Then there is the issue of measurement which is even more difficult to motivate.  In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$).  How would you convince your students that this had to be the case?  I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be.  So, what do you guys think?  Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?  Note that, at some point, you will have to make reference to experiment.  After all, this is the reason why we needed to develop quantum mechanics.  In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying.  I think we can do better.  Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory.  You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.  Edit:  To clarify, I will try to make use of a much simpler example.  In an ideal gas, if you fix the volume, then the temperature is proportional to pressure.  So we may ask "Why?".  You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.".  If you choose the latter, you can then ask why that is true.  Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example).  The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment.  And in this particular case, the objective is to do this with quantum mechanics. My husband is a teacher in New Jersey; 61 years of age. He has a pension plan listed on w-2 in box 14 (other). It is coded as 414(h). He asked the tax advise person at the school if he could still contribute maximum yearly amount to an individual Ira and was told yes. An accountant we know told us this was incorrect. Can he deduct the amount placed in individual Ira? My work has decided to issue their own  (CA) to handle different aspects of our work securely without paying for certificates.   Cryptographically sign emails Encrypt email contents Make access to things like the company  client-certificate based. Revoke the keys of former employees automatically   They sent me a .pem file, and I'm not sure how to add it to my Ubuntu install. The instructions sent were: "Double-clicking on it on a Mac should install it."&nbsp;  How do I proceed? Do I need to do something with  to create a .key, .csr, or .crt file? How to edit geometry of multiple layers simultaneously in QGIS 3? I have several line layers (route, pipes, cables) that share the same geometry and would like to edit them simultaneously.   Is there a way to easily snap the geometry of the pipes and cables layers to the route layer so that I can simply change the geometry of the route layer and don't have to repeat all changes on the other two layers as well?  To clarify, I plan a fibreglass route and have three layers. The first one describes the route that I have to take to pass all homes that have to be connected. The second one contains the pipes in which the cables lie and the third layer is the cable layer. So all features on those layers have to have the same geometry. Once I change the pathway of the route layer, I have to apply the same changes to the other two layers to ensure that they are still aligned with the route layer. I'm not sure whether I understand your comments correctly, but I use the basic editing tools in QGIS 3. Can anyone tell me if I should use inspire or inspires in this phrase?     An extraordinary leader whose vision, values, integrity and boundless curiosity inspires all who follow in his footsteps. What Hogwarts classes are taught to two houses at once? I remember from my mumble about nine mumble rereads of the Harry Potter series that the following classes Harry took were taught to more than one house in the same lesson:   Potions with the Slytherins  Herbology with the Hufflepuffs  Care of Magical Creatures with the Slytherins  broomstick riding with at least the Slytherins (maybe other houses, too?)    I don't remember any student from a house that isn't Gryffindor being mentioned in the sections on Transfiguration, Charms, Defense Against Dark Arts, History of Magic, or Divination. That would mean McGonagall had to teach twice as many classes as Snape, if both subjects got the same number of lessons (which I think I recall).   Were any subjects other than Potions, Herbology, and Care of Magical Creatures taught to more than one house in the same lesson? And do we know why some subjects are taught to more students at the same time than others? how to fill color in image in particular area? I found this , but I am not sure if this is the same issue.  On , I did some reformatting of the indentation in the code block to make it a bit more readable and it was  for peer review. A few minutes later, it appears the OP made an edit that added some additional code. It looks like my suggested edit was applied when the OP saved his/her edit, and it looks like he got the "credit" for the edit.  I'm not really concerned about the 2 rep points; I'm more curious why this happened and if this is a bug (or limitation) in the editing feature, or if this is known/intended behavior?  Note that this is not a suggested edit that's overridden by a concurrent edit: the recorded edit starts from the post-suggested edit state (and in fact can be identical to it, as in  ). Solve the integeral equation (C.S.I.R) Let $\lambda_1, \lambda_2$ be the eigen value and $f_1 , f_2$ be the coressponding eigen  functions for the homogeneous integeral equation  $$ \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) dt = 0 $$  Then   $\lambda_1 = -18 - 6 \sqrt{10} , \lambda_2 = -18 + 6 \sqrt{10}$ $\lambda_1 = -36 - 12 \sqrt{10} , \lambda_2 = -36 + 12 \sqrt{10}$ $\int_0^1f_1(x)f_2(x) dx = 1$ $\int_0^1f_1(x)f_2(x) dx = 0$   I have solved simply and get the values of $\lambda_1 \ \ and \ \ \lambda_2$ belong to (1).  Please tell me about (3) and (4) option.  Thank you. What are the differences between "=" and "<-" assignment operators in R? TeXnicCenter & Adobe Acrobat 10 How to check if a string starts with a specified string? Indefinite articles used with plural nouns: It was AN amazing TWO DAYS