I am following a course on stochastic processes. One question is to Calculate  $$lim_{n\to \infty} \sum^n_{i=0} e^{-n} \frac{n^i}{i!}$$ I see that it looks like the expected value of $e^{-n} \frac{n^{i+1}}{i!}$ but i don't know how to proceed.  I tried to use mathematica but i'm not sure this is correct.     Solve[Integrate[E^(-n) n^(i + 1)/Factorial[i], {n, 0, x}] ==      1/2 Integrate[       E^(-n) n^(i + 1)/Factorial[i], {n, 0, Infinity}] , x, Reals]   Thanks in advance
I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it.
Are there other explanations for slow tyre pressure loss? I rode my road bike on Friday and the tyres were at the correct pressure then, around 100psi.  On Tuesday, I jumped on without checking them and quickly realised something was wrong with the rear tyre.  It wasn't completely flat but was fairly soft.  I hadn't gone far so went home and took the car to work instead.  When I got home later, I was surprised to find the tyre, while still soft, hadn't lost all pressure by any means.  It was around 40psi and felt the same as when I left it in the morning.  I decided there must be a slow leak somewhere, but I couldn't find one anywhere, putting the inner under water and going around it methodically looking for bubbles.  I've never failed to find a leak before that way.  I've switched it out anyway because I don't want to mess around with my commute, but I'm curious ... is there another explanation for that amount of pressure loss?  Or must there have been a puncture somewhere that I missed somehow?  The tyres (whole bike) are pretty new and in excellent condition. Subject/Complement Agreement. How to describe problem with "The thing is the objects." In my , I provided the following, problematic,  wording (especially bold italic), and I need help to better understand this issue so I can fix my answer:1          The thing is the books. (Reduced form of Sentence 2.a)         Fundamentally, Sentence 2.a (the so-called "correct" answer), is grammatically defective. Recall there is another grammatical rule: the subject and subject's complement should match in number.    A clarification is needed here. I am not suggesting here, in this EL&amp;U question, that there is any such singular grammatical rule as indicated in the quote above.  The quote continues as follows:     The reduced sentence makes the disagreement between the subject's and complement's plurality obvious. What is one to do? Language is linear and we know "The thing is X" is better than "The thing are X", so we go with the former, which is the subject-verb agreement rule. But what is "the books" then? It must be thought of as a collective noun, even if it doesn't look or feel like one. Forcing something to be a collective noun is related to the idea of notional agreement.   Others noted that grammatically defective is too strongly worded.  There is no such "grammatical rule" for subject/complement agreement.  Please help me understand/improve upon my line of thought here.  I'm stating that Sentence 2.1 presents a fundamental "conflict" or "error" (however one might define error) at some level (morphological, syntax, grammatical, semantic):     The thing is the books.  (Or, "The thing is the objects.")   Based on the "obvious" morphosyntactic plurality conflict, the above sentence feels so wrong that I think many native speakers would think it "simply" grammatically incorrect. However, @F.E. gives counter examples:     And here are some more grammatical examples, where the number of subject and predicative complement don't agree: "They were a problem to us all", "That so-called work of art is simply four pieces of driftwood glued together" (CGEL, page 254-5). As CGEL says: "What is required is semantic compatibility, not syntactic agreement . . .". There are more examples on page 512: "The only thing we need now is some new curtains", "The major asset of the team is its world-class opening bowlers", "Our neighbors are a nuisance", "This gadget is five different tools in one".   I think all the examples given above can be explained as notional agreement.    Question 1: If we explain this phenomena in terms of notional vs syntactic agreement, where does "grammatically correct/incorrect" fit in?  Question 2: Can a simple sentence such as "The thing is the objects." be grammatically incorrect while more complex sentences such as "The thing is four pieces of driftwood glued together." be grammatically correct?  (The answer, in my mind, must be no; this is my conundrum.)  Question 3: Would it be better to say that Sentence 2.a has a low level of (linguistic) ?   Question 4: How about a low level of  and that it's semantically difficult to be (linguistically) ?    Question 5: Do we simply draw a hard line and say "The thing is the objects." is 100% grammatically correct?  I didn't particularly like this option based on intuitive notions of "grammatically correct".    1. Please help me and be kind; I'm trying to improve my understanding. I'm looking for cool, objective advice on how to look at this.  I numbered my questions so they can be easily referenced, if desired.  My preference is the final answer would sufficiently answer all the questions, but of course that can be done explicitly or implicitly. I use a brick and mortar bank for some of my savings. I’d love to get a CD interest rate closer to what Ally or other online banks offer.   Are brick and mortar banks will to negotiate with customers on interest rates? For instance, if I went into a branch and said I’m moving my money to Ally unless the bank matches Ally’s rates, do I have any chance of getting them to? How does Ubuntu partition the disk when we choose "Erase disk and install Ubuntu" during installation? If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$ Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$ Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$   I can not understand how to prove it. Please help me. How can I release and renew my DHCP lease from Terminal? From the Terminal, how can I release and renew my DHCP lease?  Currently, I'm doing the following for the wireless interface, but I'm wondering if there's a better way:  sudo ifconfig en1 down sudo ifconfig en1 up   For the wired interface, replace en1 with en0. What is the module/method used to get the current time? List of GUI Based Torrent clients Animating eye texture in a head mesh I'm trying to animate a mesh with different eye textures but it's not working.     This is a texture made by me to animate but I still couldn't figure out how to do it. I've seen tutorials on the internet but they don't work to me. When applying dropout in artificial neural networks, one needs to compensate for the fact that at training time a portion of the neurons were deactivated. To do so, there exist two common strategies:   scaling the activation at test time inverting the dropout during the training phase   The two strategies are summarized in the  below, taken from .  Which strategy is preferable, and why?    Scaling the activation at test time:    Inverting the dropout during the training phase: When to spell out numbers check the availability of the WCF Web Service I want to check the availability of the WCF web service i.c service is up or down through  the C# code. How to achieve that? How to find the shortest string in a list in Python How to fix infinite bash loop (bashrc + bash_profile) when ssh-ing into an ec2 server? I mistakenly have source .bash_profile in the bashrc file and vice versa. Now when I tried to ssh into the machine (ec2), it will stuck at loading bash and get connection closed in a second. Is there a way I could fix it? Could I mount the disk to another ec2 instance to fix the bash files?  Update 1: I tried the following: %  ssh -i &quot;my-pem.pem&quot; -t ubuntu@&lt;server_address&gt;.amazonaws.com &quot;/bin/bash --noprofile --norc&quot; Connection to &lt;server_address&gt;.amazonaws.com closed.  Nothing else showed up. Do you have any idea on what was going wrong? For sanity check, if I do ssh -i &quot;my-pem.pem&quot;ubuntu@&lt;server_address&gt;.amazonaws.com, the message will be ...  28 packages can be updated. 0 of these updates are security updates. To see these additional updates run: apt list --upgradable  New release '20.04.2 LTS' available. Run 'do-release-upgrade' to upgrade to it.   Last login: Mon Feb 22 23:17:41 2021 from ip Connection to &lt;server_address&gt;.amazonaws.com closed.  Solution Just ssh onto the machine and immediately do Ctrlc and fix the bash files. optimality of the Carnot cycle It is well-known and is stated always and everywhere that the Carnot cycle is the most efficient thermodynamic cycle. Although stated as such it is rarely proven. One proof is in the article &quot;Engine Efficiencies and the Second Law of Thermodynamics&quot;, (AMERICAN JOURNAL OF PHYSICS VOLUME 37, NUMBER 11 NOVEMBER 1969), Tobin proves that among all engines performing a cycle, so that the engine returns to its starting state, the Carnot cycle is the most efficient where efficiency is defined as $\epsilon= 1-\frac{Q_{rejected}}{Q_{absorbed}}$, and the Carnot cycle is defined as one consisting of an isothermal-adiabatic-isothermal-adiabatic segment. The proof assumes the Clausius equality, that is in a reversible cycle $\oint \frac{\delta Q}{T} = 0$. Then using the geometry of the picture below a simple argument over the areas enclosed in the cycles IABCDJI vs. IAEFGDJI (heat absorbed) or IJDAI vs. GHEAIJDG (heat rejected) will show that any other cycle working between the same temperature limits such as EFGHE will have lower efficiency than the Carnot cycle ABCDA. But this proof assumes tacitly that pieces EFG and GHE are such that $dS&gt;0$ and $dS&lt;0$, respectively. Here I am thinking of a nasty cycle such as shown in the next figure in red where the cycle cannot be neatly separated into two and only two monotonic parts having $dS&gt;0$ or $dS&lt;0$. There is no obvious reason why that restriction should hold in general. Clearly this monotonicity should hold if the engine has only temperature as its independent &quot;force&quot; but if the engine's state needs more than one independent force variable than the situation is more complicated. Instead it is reasonable to assume that any physical cycle should have finite number of monotonic pieces, kind of a &quot;thermodynamic Dirichlet condition&quot; but it could also have self intersections, little loops, and other ugly features when projected on the TS plane. How would then one prove the optimality of a Carnot cycle? Or ugly cycles on the ST plane are non-physical even if there are more than one thermodynamic variables? [] [] What USB drive filesystem works out of the box on Windows, Linux, and OS X? I've sometimes seen this construct, used to avoid repetitive handwriting, but I don't even know if it's proper English.  If it is, what is the name of this construct? I have no idea what to search for when looking it up...        Don't forget to buy eggs.   &emsp;"&emsp;&emsp;      "&emsp;&emsp;"&emsp;"&emsp;yogurt.   &emsp;"&emsp;&emsp;      "&emsp;&emsp;"&emsp;"&emsp;milk. Is there a formal spelling for the English letter names?