Prove by induction that for any $n \ge 2$ :  $$\sum_{k=n+1}^{2n}\frac{1}{k} \ge \frac{13}{24}$$  I obviously know the base case is true, but beyond that it seems to get convoluted and I'm not sure how to solve this question.  Anyone help?
Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} &gt; \frac{13}{24}$ for $n&gt;1,n\in\Bbb N$  To solve it I used induction but it is leading me nowhere my attempt was as follows:  Lets assume the inequality is true for $n = k$ then we need to prove that it is true for $k+1$  so we need to prove $\frac1{k+2} + \frac1{k+3}+\cdots+\frac1{2(k+1)} &gt; 13/24$  I don't know where to go from here please help.
How do I "parent" objects? I'm looking for a way to have two separate objects be connected, so that when I move the first object the second one will move also. The term parent is used, but how do I parent a object? My wife wants a 4th baby, but I don't.  We have been talking about it for the last few days (talking strongly).   I have raised points such as:   We will get less time to spend with our current kids. The age gap between the first and the last will be too wide. It will cost more money. She will be out of work longer. We will need a bigger house. And we don't even have our own house yet (we are still renting). We will have to start again. Sleepless nights, nappies, feeding, new cot, car seat, etc etc.   She says:   She feels like she is missing something. She has always wanted a large family. Things will be the same regardless of having 3, or 4 kids. She agrees that money will be tighter, but wants to push through it.   Our current kids are 9 months, 2 years, and 4 years. We are young married couple (25), and I have a fairly decent job.  What would be the pros and cons of having another child?   How can we rationally talk to the other about having/or not having another child?  And what should we do if we cannot agree? Inequality from Chapter 5 of the book *How to Think Like a Mathematician* This is from the book How to think like a Mathematician,  How can I prove the inequality $$\sqrt[\large 7]{7!} &lt; \sqrt[\large 8]{8!}$$  without complicated calculus? I tried and finally obtained just $$\frac 17 \cdot \ln(7!) &lt; \frac 18 \cdot \ln(8!)$$ Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$ How to get a combined ticket in IRCTC (Indian Railway) Suppose I want to go from Station A to Station C but Station C is not a stoppage of some train say train 01, so I can get a combined ticket from station A to Station B  through train 01 plus Station B to station C through a different train say 02, so my total fare will be of same ticket as from Station A to Station C.  We can get Combined ticket from the Booking Window but how's that possible on the IRCTC website? Azure DNS does not lookup SPF policies for certain domains Run VirtualBox in background, without a window? I was wondering if it's possible to run an Ubuntu virtual machine using VirtualBox in background, I mean, without any window open.  The idea is to connect via SSH to the Linux host, that would be running in background.  Has anyone ever done something like this? Is it possible? What is the difference between independent and mutually exclusive events? Two events are mutually exclusive if they can't both happen.  Independent events are events where knowledge of the probability of one doesn't change the probability of the other.  Are these definitions correct? If possible, please give more than one example and counterexample. Evaluating and negotiating a start-up package I've been offered an academic position, and am now in the fortunate, but unfamiliar position of negotiating the offer.  I've seen a few blogs/comments/posts on this process.  However, I have no real idea how to evaluate the offer that's been made.  Coming from the position of a postdoc, the package seems impressive (given that, as a postdoc, my "package" consists of a very modest salary and health benefits).  However, I also realize the importance of maximizing (within reason) my initial offer, as this sets the baseline for, at least, the first few years of my faculty career and could provide support that would prove very important in the initial year or so while I'm trying to obtain grant funding.  In particular, I'm looking for two types of information.  First, what types of startup packages are standard for tenure-track assistant professors of Computer Science at a public RU/VH (i.e. Research) university?  This is just to give me some point of comparison.  Second, what types of changes are reasonable to negotiate?  Can I ask for more summer salary support while still conveying to them my desire to obtain grant funding as quickly as possible?  The offer already includes a reduced teaching load for the first few years --- is it also reasonable to ask for an extra semester off prior to a tenure review?  I understand these are fairly broad questions, but any input or suggestions would be very helpful.  I'm definitely interested in the job; I really liked the department and think it's a great opportunity.  I don't want to alienate anyone by pushing too hard, but at the same time, I want to make sure that I make every reasonable effort to maximize my chances for academic success, which includes maximizing my available resources by negotiating the best startup package I can. How to compare Unicode strings in Javascript? Whenever I'm running or playing a game, I consistently run into an issue of setting atmosphere. I've realized the best way to do this is through more effective, engaging descriptions of settings, actions, and objects that the characters see and feel.  As a player, I feel that my actions are generally "I attack the goblin," or some variation thereof, but I'm afraid that I'll stumble over my words and further confuse the other players and I'll resort to "I attack the goblin" as to not drag the game out longer.  As a DM, my issue tends to be more with setting atmosphere with settings, locations, and objects. Effectively, the flavor texts that comes with walking into some amazing location that the characters (not necessarily the players) should be suitably impressed by. My goal here is to engage the players with some degree of spectacle, and be more inquisitive in their environment and whatever lies within it.  How can I create more descriptive, concise flavor text for the games I play in? Does 10 heads in a row increase the chance of the next toss being a tail? I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of probability and/or statistical jargon is tossed around (excuse the puns).   Assuming that is the case, my question is this: how the hell do I convince someone that is the case?   They are smart and educated but seem determined not to consider that I might be in the right on this (argument). What is the command to open the shutdown dialog? A saying for something that's good but also has a downside I'm trying to transition from a list of pros to a list of cons, and would like to have some sort of transition quote. Something in the form of,  "But, alas, (((something something quote quote)))"  Everything that has a beginning has an end, something along the lines of that. Does a multiple-entry Schengen visa allow visiting other Schengen countries later, without going via the issuing country? What is the final price if the bid price exceeds the ask price in an order driven market? Why can Naruto still use Rasenshuriken? Word for killing off or attempting to kill off an entire bloodline? I would like to know if there is a word or phrase for a systematic campaign to annihilate every descendant of a particular ancestor.   We have the word 'genocide' for killing off an entire race, and plenty of words for killing people based on how they are related to you (fratricide, filicide, et. al.), but looking over the  and the , I couldn't find a term for systematically killing an entire bloodline.   My question is prompted by a US news story about the horrific murder of 8 members of the same family, but those murders are not an example of what I'm thinking of because the children of the victims were left alive, so the motive for the killings was not to kill off the bloodline. The sense that some group had coordinated an attack against just one family is what made me wonder what the "not an entire race, just one bloodline" version of 'genocide' would be called.   Searching the Internet for "kill off" and bloodline sends me straight into the land of vampires, Illuminati, reptilian overlords, and, naturally, Priory of Sion nonsense so I haven't had much luck finding a thread to pull.   It may be that there is no word for it in the sense of murdering a human bloodline, because we just don't attempt kill off our royals down to the last heir often enough to coin a term for it. Maybe there is a scientific term for culling a line from a test population as part of biological research? Return a blank when cell reference is blank I am using the following formula in a Google Sheets to calculate the number of days between two dates (dates are in columns G and AL):  =ARRAYFORMULA(if(row(A:A)=1, "Readmission Days", (G:G)-(AL:AL)))   It has to be an array because I am working with a Google Forms and entries are automatically populated into new rows.  The problem is that some of the cells in col AL are blank, so the formula is reading those dates as something else and returning a very large number. Also, when both cells are blank, the formula returns a 0, messing up my data.  How can I include something to ignore blank cells or just return a blank when one or both cell references is blank? RegEx: Grabbing values between quotation marks