Let $x_n \colon= \sqrt[n]{n!} $ for all $n \in \mathbb{N}$.   Then how to determine rigorously whether the sequence $\left( x_n \right)_{n \in \mathbb{N} }$ converges or diverges?   And, how to find $\lim_{n \to \infty} x_n$, rigorously?  By rigorously I mean using the same machinery as has been developed by Rudin until Chap. 3, where he discusses sequences.   My Attempt:      We note that, for all $n \in \mathbb{N}$ such that $n &gt; 1$, we have   $$ 1 \leq \left( x_n \right)^n \leq n^{n-1}, $$   and so    $$ 1 \leq x_n \leq n^{ (n-1)/n } = \frac{ n }{\sqrt[n]{n} }. $$   However, although    $$ \lim_{n \to \infty} \sqrt[n]{n} = 1, $$   we also have    $$ \lim_{n \to \infty} n = +\infty. $$   Thus the squeeze theorem is not applicable.    Or, can we find some majorizing sequence converging to $1$?  Is this sequence monotonic?
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
Is it essential to use comma in a big number? Is it necessary to use comma in a big number? For example, if anyone writes one million four hundred thousand and two as 1400002 instead of 1,400,002 on answer sheet of listening section of IELTS test, will it be considered incorrect?  Thanks in advance. New to the cloud so pardon this question as I'm certain it's relatively simple!  I've setup my Google Cloud Instance and I'm now trying to connect to it using FileZilla. I have a good understanding of how to do this... but I've been completely unable to download my .ppk file, public key, and private keys. The documentation clearly gives the folder that the files are located in.... but I'm unable to navigate to these folders (it's as if they don't exist) furthermore I'm not really sure how to download something to my computer through the SSH terminal.  Also somewhat infuriating is that I've found the keys in the web portal on Google Cloud however there is no option to download them. I can only view them in clear text.  Any help would be greatly appreciated! Objects stuck in solid mode? I can't figure out what is causing this I tried removing the materials for the objects as well as re-assigning new materials and flipping normals I don't know what is causing this issue for the model.  The head seems to be working like it should but the body doesn't turn white in textured mode or black in material mode. I'm using default blender render I'm using Android 4.4.2 (on ). I've a Windows PC at home which has highspeed broadband internet connection. I don't have wireless router or access point. I just want to use PC's internet connection on my phone. I've tried multiple methods but none seems to work.  Method 1: Using native Android 4.4.2 option  In my phone when I connect my phone, in Settings &gt; Wireless &amp; networks &gt; USB Internet (sharing Windows PC internet via USB cable) checkbox, which I checked then it asked me to choose my Windows version (which is Windows 8.1 but since it didn't show that option) I chose Windows 8.  It gives me further instructions as showing in the following picture:    Here is the problem, there is no sharing tab (see the following screen shot)   Method 2: As described in   I learned on this site that it is called Reverse tethering. I read and followed the procedure mentioned in    I followed all the steps described in that answer, until executing the command.   Install USB drivers from Android SDK [DONE] Connect USB cable and activate USB Tethering (You should see on a new network interface.) [DONE] Bridge the 2 network interfaces [DONE] On your computer execute adb shell netcfg usb0 dhcp [PROBLEM]   When I execute adb shell netcfg usb0 dhcp it says: error: device not found. I tried the same thing with USB Debugging in Developer settings, ON &amp; OFF. The error remains the same.  Method 3: Rooting the device and using an app  I read here and there that there is another method, in which we can directly use an app but it requires rooting the device. My phone is a new phone and warranty becomes void if I root it.   EDIT: I've tried the method suggested in answer, at the 4th step of method 2, I'm facing this problem: Will my checked luggage freeze in the airplane? I'm taking some freeze-sensitive items in a checked luggage for a short flight. It's cheese, chocolate, toothpaste and wine. I would like to know whether the luggage can get "frozen" in the aircraft. Sometimes it happened to me that my stuff actually was quite cold after the flight, so I'd like to be quite sure. Especially the cheese was really expensive and freezing it would very likely demage it.  Usually I pack everything well in clothes, but this time the things are just too many so it's tough to do more than just protect the wine from breaking.  Some details: It's an A320 flight taking 1.5 hours, with Czech Airlines, from Paris CDG to Prague PRG. Ubuntu 14.04, Intel Chromebook, no sound How can I safely create a nested directory in Python? Find the value of : $\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$ LED as power generator Prove that the product of four consecutive positive integers plus one is a perfect square I need to prove the following, but I am not able to do it. This is not homework, nor something related to research, but rather something that came up in preparation for an exam.     If $n = 1 + m$, where $m$ is the product of four consecutive positive   integers, prove that $n$ is a perfect square.   Now since $m = p(p+1)(p+2)(p+3)$;     $p = 0, n = 1$ - Perfect Square      $p = 1, n = 25$ - Perfect Square      $p = 2, n = 121$ - Perfect Square   Is there any way to prove the above without induction? My approach was to expand $m = p(p+1)(p+2)(p+3)$ into a 4th degree equation, and then try proving that $n = m + 1$ is a perfect square, but I wasn't able to do it. Any idea if it is possible? Cleaning Contaminated Disc Rotors I've had an eternal struggle with my current bike over brake contamination. The levers and calipers are shimano ultegra r8000, so the latest and (almost) greatest.  I first noticed it when I took it to a repair shop (i no longer use) for a bleed, it came back and the front brake would squeal like crazy and at low speeds had little to no braking force. You can stop from speed but then struggle to come to a halt without using your foot.  At around 2000 miles I tried to replace the brake pads, when I took them out they were black with grime on both sets (front &amp; back), not to mention worn down to the metal. I replaced them with the L04C sintered pads, gave the callipers a good cleaning with a towel (no chemicals) and fitted the new set.  Annoyingly all of my braking issues are back, now I probably should have cleaned the rotors, I simply used some muc off disc brake cleaner on them. I would like to know the best way to clean the rotors, I've inspected the pads and they are looking fairly black now, with plenty of tread left. I'm going to abandon these pads and fit a new set. For the rotors I plan to completely remove them from the wheels (they are center lock fitted) and try and treat them off the bike, what is the best way to fully clean them? They have plenty of wear left in them but you can see these black grooves in them, and plenty of muck on the back that I struggle to get off.  Below are images of the rotors after soaking them in disc brake cleaners, waiting and then wiping them down with a clean, microfiber towel.   Front Wheel Outside Rotor   Front Wheel Inside Rotor - note the increased dirt on this side  Looking at ways people clean their disc rotors, I've seen plenty of suggestions that I would need to burn the contaminants off the discs. Would the best approach be to remove them and use a blowtorch, should I soak them in some nasty chemicals overnight before I try to heat them? Do I try baking them in the oven?  And finally, what is the best way to prevent the brakes from becoming contaminated in the future? it's really off putting when they squeal and don't perform how I'd expect because it means I'm more reserved whilst out riding on the bike, leading to reduced time riding around outside. I commute on this bike (50 miles a week) and then ride about 50 on the weekends also, so it see's it's fair share of road use. I tend to wash it once after my big ride on saturdays and will rinse it off if it's been raining, any day that occurs (since I live in north east england, the acid rain..). I've owned disc brake bikes before and never had half the issues I've had with this, I had a cheap trek with tektro brakes and the brakes were sharper and never had any issues with contamination whatsoever (i did around 500 miles on that bike before a motorist happened).  Thanks for any and all advice :) How do Prestige Classes that advance spellcasting interact with racial spellcasting ability? Do DOM tree elements with ids become global variables? Suppose that G is a group and N &lt; G a normal subgroup. Assume that there is no normal subgroup M &lt; G with N &lt; M. Prove that G/N is simple.  This problem is really throwing me for a loop. The concept of quotient groups and how they relate to simple subgroups is confusing me, and so I am not really sure how to begin. If a simple subgroup's only normal subgroup is itself (and the trivial group), then G/N being simple would be proving that when multiplying the entire subgroup N by an element g for all g in G, that the collection of those sets cannot be broken into smaller normal subgroups? Should I always use a comma after "e.g." or "i.e."? It seems that "e.g." is always followed by a comma but "i.e." is not. Why is that? Should you pick a baby up everytime it cries? My husband and i have differing opinions on this. I don't believe in the term 'spoilt'. I would be very happy if my daughter was 'spoilt' in the sense that she expects to be treated with the highest possible respect and love and affection in all her relationships.   Also it would not bother me if she think she is special and unique (that's exactly what I want her to think). Mother's face is a mirror for the baby - through the mother's reactions and attention, the child learns that she is a lovable and valuable human being.   I spend all my time with my baby when she is awake. I play games with her, I smile, and laugh, and I never ever let her cry. I rush to pick her up immediately, even if it's the middle of the night, I force myself to get up, no matter how tired I am (I admit it's hard sometimes, and I do let her cry for 10-20 seconds sometimes, but never more). How does the propagation of gravity work for photons? What is the weight equation through general relativity? The gravitational force on your body, called your weight, pushes you down onto the floor. $$W=mg$$ So, what is the weight equation through general relativity? I found  where a technician with a lot of experience seems to clean his PCBs using water, and then he just puts it in the sun so it can dry. I was just amazed to see the circuit keeps working as intended, he says he keeps doing this all the time and several times, however I doubt this is normal.  Which leave me some questions:   What are the recommended methods to clean huge PCBs like the one in the video? What happens to charged capacitors when water shorts them out? Solder joints do not get any kind of corrosion? Is the antisolder mask waterproof? When changing fields, should I focus more on describing past research or connecting it to the new area in my statement of purpose? I want to change my field from molecular biophysics to theoretical ecology. While they are basically required some same basic background (physics, math, programming), the rest is quite irrelevant. In my current draft of this part, about a half is about this the problem and the result of the work, another half is for my contribution and what I've learnt. Having said that, should I focus less on describing the work and leave space for the thing that will be useful for the next research?  Below is copy from .     What have you already done? What problems have you solved, or at least worked on? What independent projects have you been part of? What were your key contributions? What did you learn? What did you teach the world? How do your results compare to what was already known? What original ideas are you most proud of? Be specific, technical, credible, and confident (but not arrogant). Refer the reader to your web page for more details. Have a web page with more details: preprints, project reports, source code, videos, etc.