@DHMO That could be an interesting question. I don't really know, but it's easy to see that same power series expansion extends as an entire function on all of C. But the fact that it's related to cos(z) and sin(z) is a formal fact I don't know how to intuitively describe
OK, @DHMO. So you want to understand complex analytic functions as complex differentiable? That's in general not a great intuition unless you know the equivalence with them having power series.
That's the whole point of complex analysis, like Ted said
Suppose you have an analytic function that's zero at an infinite set of points converging to $0$. You can prove (pretty easily) that all the derivatives at $0$ must be $0$.
No, there are plenty of continuous functions with zeroes that accumulate.
@TedShifrin Just double checking: is the disk pi*integral of (2-\sqrt{5-(y-1)^2})^2) from 2 to 1+\sqrt{5}? I'm assuming that the integral bounds are the same for both the washer and disk, and we find those by taking [0,2] for x and then solving for y
@DHMO Anyway, the thing is power series are intuitive. It is however a potentially interesting question how to read off periodicity from power series, or an infinite product, or something. I don't know.
But just because it's not intuitive to understand one single property from it doesn't mean it's worthless :)
@DHMO: Derivatives of $e^{-1/x^2}$ (with $0$ at $0$) are of the form $P(1/x)e^{-1/x^2}$, where the degree of the polynomial $P$ is quite predictable. So these approach $0$ as $x\to 0$. If I fix $x\ne 0$, they do approach $\infty$ as $n\to\infty$.
Also the game is more aware of things than one would expect. Actions such as resetting are in game, so some characters kind of know what's going on, and adjust how they interact with you accordingly
@TedShifrin Thanks for the help--I have a quick follow up question. Same circle, but the interval for x is now [1,3], and I am revolving this new region about y = -2. If I do the washer method, I think the big radius is 2, but I'm not sure what the smaller radius is
Well, if only to push it away from geometry/topology (and also because you're here @Ted), do you have any hints regarding why if a sequence in $\ell^p$ for $1 < p < \infty$ converges weakly, and the norms converge, that it converges strongly?
@TedShifrin Sorry to bother you again! Same circle, but [2,4]. I am supposed to find the surface area of the solid generated by revolving this region about x = -1, so I think I take dy for this. x = 2-\sqrt{5-(y-1)^2}. Inside the integral, I have x * \sqrt{1+(dy/dx)^2} dy? Thanks a lot!
I know that $(\ell^p)^* \simeq \ell^q$ where $\frac{1}{p} + \frac{1}{q} = 1$, and that weak convergence in those spaces is equivalent to component-wise convergence
albeit we didn't have time to get things aligned before the deadline, i.e. have a registered video convertor and editor to avoid obtrusive stamping, and ensure a decent render quality. pretty fundamental stuff tbh ://
@TedShifrin Late update, but got it! Drawing a picture helps :) ... if I had to find the area under Archimedean spiral r = theta from theta= 7pi/4 to theta = 3pi, would I have to account for overlap, or is simply taking A = 1/2 of (integral of theta from 7pi/4 to 3pi) enough?
Hello, in my notes I have written (repeatedly), $F=Hom_{R}(M,-) : Mod-R \rightarrow Mod-R$ - in what way does this make sense? I don't see how the morphisms between two $R$ modules form a $R$ module.
I don't see why it's not $F=Hom_{R}(M,-) : Mod-R \rightarrow$ (Abelian groups).
Two workers A and B are engaged to do a piece of work. A working alone would take 8 hours move to complete the work that when work together.If b worked alone ,would take 4 1/2 hours morethan when work together.Time required to finish the work together is
though I would note I'm only inspecting the figure, which isn't necessarily precise or to scale
Suppose a straight line connects the origin to the particle as it traces out its path, between B and D, what happens to the angle between the positivie $x$-axis and this line segment? @MathisLife
It fascinates me to know that there are only countably many definable real numbers, precisely because we only have countably many finite strings to define a number.
@BAYMAX the first part just means that $A$ is measurable and there is an $n\in\Bbb N$ such that the third line is true. Ascending collection means $E_1\subseteq E_2\subseteq E_3\subseteq E_4\cdotd$. For the last one "$E$ has finite measure" is part of the theorem
in the question, they ask us to prove that a line segment joining the point of contact of 2 parallel tangents to each other is the diameter of the circle
@LarryFreeman Good morning, I believe that you can find a version from arXiv: Sondow, Ramanujan Primes and Bertrand's Postulate, The American Mathematical Monthly Vol. 116, No. 7 (2009), pp. 630-635. I am saying it because I see in your profile your that you are interested in it. Is not required a response of this comment, good luck.
Friends
Just came across an excercise that asks to show that set of numbers in $[0,1]$ which possess decimal expansion not containing the digit $5$ has measure zero ?
How do i approach this ?
I tried thinking of relating this to Cantor set but , i see that numbers whose decimal expansion conta...
Hi @KanwaljitSingh your method is correct, but i have a doubt on regarding on this
But book answer is 1/(x+8) + 1/(x+9/2) = 1/x , i have a doubt regarding on this x be the piece of work, then 1 hour it will be 1/x , but it will take time morethan 8 hours means we have to add like this only - (1/x ) + 8 but how they are adding like 1/x+8 please clarify , x work means how they add value 8 with that
If $\phi$ is an atomic formula; is it possible for a variable $v_i$ to occur free in it? I assumed it was not but now I have seen it spoken of in a pdf
@Learninguser your icon is so pale it appears like you have been abscent in the chat for a while even after you just posted.
How about this @DHMO since image of $f$ will be an interval or union of intervals and intervals are measurable and so f is also measurable , but here i have never assumed that f is monotone?
Theorem 7
In the question, they ask us to prove that a line segment joining the point of contact of 2 parallel tangents to each other is the diameter of the circle
but
then in the proof
they say angle CAO is a right angle.
so they are saying that AO is the radius
but they still havent proved i...
Therefore degree of variable x in minimal polynomial of T will have the same parity. If minimal polynomial of T is q and deg q< n, q(x)q(x)=P'(x^2) and p' is an annihilating polynomial of T^2 having deg<n which is contradiction
Friends
Just came across an excercise that asks to show that set of numbers in $[0,1]$ which possess decimal expansion not containing the digit $5$ has measure zero ?
How do i approach this ?
I tried thinking of relating this to Cantor set but , i see that numbers whose decimal expansion conta...
Theorem 7
In the question, they ask us to prove that a line segment joining the point of contact of 2 parallel tangents to each other is the diameter of the circle
but
then in the proof
they say angle CAO is a right angle.
so they are saying that AO is the radius
but they still havent proved i...
