Topology of $\mathbb{R}^{\infty}$
Definition of $\mathbb{R}^\infty$
Creating unnumbered chapters/sections (plus adding them to the ToC and/or header) I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar good upper bounds for $\binom{n}{k}$?  Edit: As the link in Macavity's comment shows, the bound is indeed $\binom{n}{k}\leq \left(ne/k\right)^k$. How can we prove this? Is there any difference between Present Perfect and Present Perfect Continuous? I have studied "present perfect" and "present perfect continuous" for a week. I know forms, verb and helping verb I should use when I write them.  For me, they have nearly same definition because I can use them interchangeably in Thai language. (My native language)  For example, the sentences        I have learned English language in the past few weeks.    I have been learning English language in the past few weeks.       have no difference in meaning to me. They both mean "I began learning the English language in the past and I am still learning it"  Can I use them interchangeably in English? Or is there any difference between them? For example, when should I use present perfect but not present perfect continuous? Predict probabilities from Firth logistic regression in R I'm using the logistf package in R to perform Firth logistic regression on an unbalanced dataset. I have a logistf object:  fit = logistf(a~b)   Is there a predict() function like on that's used in the lm class to predict probabilities for future data points? Or do I have to manually input the estimated parameters from the Firth regression. Prove $n!>2^n$ for $n\geq4$ using induction. I just want to know if my proof to this question is correct. First, I proved it was true for $n = 4$. $$4!&gt;2^4$$ $$24&gt;16$$ Then, I assumed that it was true for $n=k$. $$k!&gt;2^k$$ Afterwards, I tried to prove it for $n=k+1$. $$(k+1)!&gt;2^{k+1}$$ $$(k+1)k!&gt;(2)(2^k)$$ Since $k!&gt;2^k$, am I allowed to simply do $$(k+1)(2^k)&gt;(2)(2^k)$$ $$(k+1)&gt;2$$ and this is true for all $k\geq3$. Would the last two steps be illegal because I only assumed that $k!&gt;2^k$ is true? How to convert a currency string to a double with jQuery or Javascript? .NET offers a generic list container whose performance is almost identical (see Performance of Arrays vs. Lists question). However they are quite different in initialization.   Arrays are very easy to initialize with a default value, and by definition they already have certain size:  string[] Ar = new string[10];   Which allows one to safely assign random items, say:  Ar[5]="hello";   with list things are more tricky. I can see two ways of doing the same initialization, neither of which is what you would call elegant:  List&lt;string&gt; L = new List&lt;string&gt;(10); for (int i=0;i&lt;10;i++) L.Add(null);   or  string[] Ar = new string[10]; List&lt;string&gt; L = new List&lt;string&gt;(Ar);   What would be a cleaner way?  EDIT: The answers so far refer to capacity, which is something else than pre-populating a list. For example, on a list just created with a capacity of 10, one cannot do L[2]="somevalue"  EDIT 2: People wonder why I want to use lists this way, as it is not the way they are intended to be used. I can see two reasons:   One could quite convincingly argue that lists are the "next generation" arrays, adding flexibility with almost no penalty. Therefore one should use them by default. I'm pointing out they might not be as easy to initialize. What I'm currently writing is a base class offering default functionality as part of a bigger framework. In the default functionality I offer, the size of the List is known in advanced and therefore I could have used an array. However, I want to offer any base class the chance to dynamically extend it and therefore I opt for a list. What is the correct formula for gravitational time dilation for a satellite in a circular orbit? Can't transfer purchases from iPhone to iTunes in iOS 9 Why did the Resistance automatically assume the use of this tech in The Last Jedi? Is it possible to install Steam games to two different locations? I want to build a new PC with a smallish SSD.  I'd like to be able to install some of my games onto the drive (those I care about loading quickly or I play frequently), but have the rest of them on a larger, slower drive.  Anybody know if this is possible?  I'm kind of guessing it's not, but figured I'd ask anyway. Fallout Shelter :: Does Mr. Handy Pick Up Loot? I know that Mr. Handy will pick up caps when on walkabout; will he also pick up weapons and/or outfits? How to preserve custom number formatting after closing a workbook? Which Harry Potter works are considered canon? ffmpeg - Create a video from image frame with a start and a count My sister, cousins, and other relatives speculate that my mom's grandmother was Jewish (for various reasons). I've heard people say at times things like, for example, "I'm 50% German" or "I'm 25% Swedish." This has to do with their ancestry. But, how exactly is this calculated? Say for example that my mother's paternal grandmother was indeed 100% Jewish, what would that make me and my siblings, and how is it calculated? Testing points in PCB How to find normal and lognormal moments, given partial information? $Y=\ln(X)$. $X$ is lognormal and $Y$ is normal. If all I know is the arithmetic mean of $Y$ and the standard deviation of $X$. What is the formula to calculate the arithmetic mean of $X$ and the standard deviation of $Y$? Opening Raster stored in Esri Geodatabase using QGIS Is there a way to open a raster that is stored in ESRI V10.0 Geodatabase in QGIS V2.4?    I can open the vector layer, but there does not seems to be a way to open a raster. I put 4 different layers into my ArcMap document, but only one of them will show up on the map! I should be seeing the other 3 layers as well. When I do a Zoom to layer, I can see one layer, but I cannot figure out how to make the other 3 show up at once on top of each other.  Before I opened ArcMap, I used ArcCatalog where I right-clicked 3/4 of the shapefiles and  clicked Properties &rarr; Coordinate system &rarr; Import and then clicked on my Trees layer, which has the coordinate system I want the other 3 to have. But when I opened ArcMap the 4 layers did not overlap, nor could I move them above or below each other.