For each of the following, give an example of a sequence of functions $f_n(x)$ that converges to f  A. uniformly but not in the mean square sense.  B. in the mean square sense but pointwise nowhere.  I know that for part A the domain of the function cannot be bounded. Please don't just repeat the definition of what it means to converge uniformly, in the mean square sense or pointwise.  I'm looking for clues on how to choose a $f_n(x)$ that matches the conditions
For each of the following, give an example of a sequence of functions $f_n$ that converges to f  A. uniformly but not in the mean square sense.  B. in the mean square sense but pointwise nowhere.  I know that for part A the domain of the function cannot be bounded but other then that I'm not sure how to go about the problem
How are Contacts in Marketing Cloud counted? Salesforce bills Marketing Cloud on total number of Contacts within the system. I'd like to know what exactly is used to count these? Is this the Email Studio > All Subscribers list or the Contact Builder > All Contacts list?  When we have Marketing Cloud connector and synchronise the Contacts object, is this counted towards that number? (or only when they have been inserted into the All Subscribers / All Contacts list, as they are not done so automatically) Weight Painting Won't Get Rid of Weird Protrusions How can I get back to Blender's default theme? Standard purpose of favorites I am wondering if my behavior regarding favorites is typical.  More importantly, if it isn't typical, does my behavior conflict with an internal metric designed to reward favorite'd questions?  I mark a question as a favorite for a few reasons:  A -- I find the question/answer(s) remarkable enough that coming back to them time and time again is of benefit to me.  B -- I am waiting to see if a sub-par quality question is going to be revised enough for me to come back and answer it.  Either way, I eventually remove my favorite.  C -- I have answered a question (of any quality) and am wanting to be notified of any competing answers.    Situations B and C are the most common for me and they typically mean I will go back and remove my favorite star later; I don't remove it because the quality of the question degraded, I merely don't want to be notified any longer.  Even stranger, I am often favoriting questions that are rather low-quality (reason B) making me wonder if low-quality posts are being rewarded via some invisible metric because I (at least temporarily) have given it what looks like a stamp of approval akin to an up-vote.    I suppose the feature being called "Favorite" makes me confused because most of the questions I select are not my favorite.  Am I accidentally messing with an invisible metric that is drawing attention to posts when that isn't desired behavior? Stirling's Approximation Proof I was searching for the reason why Stirling's Approximation holds true. I found the website  which apparently shows why this is the case. Does this part of the equation make sense in the proof? If so why?  $$\int_{1}^{n}\ln(x)dx\approx\sum_{k=1}^{n}\ln(k)$$ How to sort a list of strings? What is the best way of creating an alphabetically sorted list in Python? Is there a term for grammatical mistakes as a result of trying too hard? Layers with same coordinates not lining up I have a project where I have a few layers (State of Texas, cities, roads, etc.). They are all resaved and projected to NAD83 / UTM zone 14N EPSG: 26914. The project CRS is also NAD83 / UTM zone 14N EPSG: 26914 and on the fly projection is enabled. I added a text delimited file and cannot get them to line up. I checked the attribute table of the delimited file and the coordinates are correct. I cross-checked this with a point close to where a few of the points from the delimited file are and they are very close so in theory they should be lining up. I tried to save the delimited file and save it with the same projection and add the shp file back in and that did not work. Anyone have any ideas? I can't think of anything else to try. The characteristic and minimal polynomial of a companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.  It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are both equal to $f$. This can be seen quite easily by induction on the degree of $f$.  Does anyone know a different proof of this fact? I would love to see a graph theoretic proof or a non inductive algebraic proof, but I would be happy with anything that makes it seem like more than a coincidence! In which way does quantum mechanics disprove determinism? I've heard this pop up in a discussion with my physicist/engineer roommates, but didn't care to ask at the time.  Now I'm mighty curious about it.   doesn't really seem to say much on this issue.  From what I understand about the Uncertainty Principle, it says that there are certain properties of electrons and stuff that cannot be measured, and are therefore uncertain.  Then Wikipedia (under ) states that Sir Arthur Eddington says that the Uncertainty Principle isn't really so because we can't measure these properties, but because turns out nature is indeterministic.  At least that's what I took from those paragraphs.  Even without my biased wording, it sounds more like an assertion than evidence.  I've also read a few things about how other scientific conventions perceive the issue, like how a ball on the peak of a perfect mound might randomly roll down in any direction, and I'm still unconvinced.  My belief of determinism is generally that if you knew every single variable that existed as a factor at the very beginning and birth of the universe, you could correctly determine all properties of any individual particle at any point in time.  Could anyone provide some more background about this?  Especially regarding quantum mechanics? Is there any general and systematic way of reducing the higher order linear ODE to a system of first order ODE?  For example, assume we have $a_3x^{(3)}+a_2x^{(2)}+a_1x^{(1)}+a_0x=0$, then how do we convert this into matrix form(a system of first order ODE). And after we solve the system of equation, how to combine them into our final solution $x(t)$?  Thanks for helping me out! Every Christmas, Harry receives a mediocre gift from the Dursleys. In Harry Potter and the Chamber of Secrets he gets a toothpick.     At that moment, Hedwig swooped into the room, carrying a very small package in her beak.     “Hello,” said Harry happily as she landed on his bed. “Are you speaking to me again?”     She nibbled his ear in an affectionate sort of way, which was a far better present than the one that    she had brought him, which turned out to be from the Dursleys. They had sent Harry a toothpick    and a note telling him to find out whether he’d be able to stay at Hogwarts for the summer    vacation, too.    They hate owls, but obviously use them to contact Harry at school? Printing the value of a variable in SQL Developer Dark Mode for Stack Exchange sites Each Stack Exchange site should provide users the option of using a dark mode design/theme.  Why is this feature missing?  The major IDEs are already a letdown regarding dark color schemes. Well the question is to solve for $x$ in $$a^x+b^x=c \tag{a,b,c are constants}$$   Well as of me, I tried to put $\ln{}$ on both sides which does not seem to help. Apart from this I don't seem to have any other way to solve it.   I turned to WolframAlpha and the solution which was given was : $$x=\dfrac{-i(2\pi n +\pi)}{\ln{a}-\ln{b}} , {n \in \mathbb{Z}}$$  And if $n=0$ $$x=\dfrac{-i\pi}{\ln{a}-\ln{b}}$$  The only thing I am able to recognize is $e^{i\pi}=1$ ... i guess is used.  Link :   So I see that $x$ is independent of the constant $c$.  So how to solve it? Please help...Thanks!    EDIT - It seems that WolframAlpha is taking $c=0$ as suggested by Donkey_2009. And when i write "Solve for x in a^x + b^x = k" it is not returning anything. So are you guys aware of a method to solve this? Also are you guys aware of any other website/software like WA?    P.S. - Also I see that IF we know the values of $a,b,c$ resulting in something like $2^x+3^x=100$ , then Newton's method is the only method I am aware of that can solve it...  Any other suggestions? Note: This is an attempt to create a canonical question that covers all instances of &quot;low-graphics mode&quot; error that occurs to a user, including but not limited to installation of wrong drivers, incorrect or invalid lightdm greeters, low disk space, incorrect installation of graphics card like ATI and Nvidia, incorrect configuration of xorg.conf file while setting up multiple monitors among others. If you are experiencing the &quot;low-graphics mode&quot; error when trying to login but none of the following answers work for you, please do ask a new question and then update the answers of this canonical question as and when your new question gets answered.   When I try to boot into my computer, I am getting this error:  The system is running in low-graphics mode Your screen, graphics cards, and input device settings could not be detected correctly. You will need to configure these yourself.   How do I fix the failsafe X mode and login into my computer?  Answer index: Why exactly does diffraction occur? Why do waves that were traveling in a straight direction change direction when passing through an opening?  I thought that the waves (red arrow) when colliding with the wall bounce in the opposite direction (green arrow).  And the waves that pass through the aperture follow its path normally as shown in the image on the right.   The waves that go in a straight direction should follow traveling straight line like a car that goes under a bridge the car is straight on the road. But this is not so.  Why does the direction of the waves change?  How is the direction of the waves calculated? Why is radian so common in maths? I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)  My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some intrinsic advantage over degrees?  For me personally it doesn't matter if I write $\cos(360°)$ or $\cos(2\pi)$. Both equals 1, so why bother with two conventions? I read somewhere that companies are not required to pay dividends to shareholders (this is correct, yes?).   If so, then if company A never pays dividends to its shareholders, then what is the point of owning company A's stock? Surely the right to one vote for company A's Board can't be that valuable.   What is it that I'm missing? x tick label style issue, anchor=east and text width command not working together