Thank you for your advice. I have Apostol's analysis book. And yes it seems Apostol's book covers measure theory and Lebesgue integration, but I heard it's somewhat limited in its content compared to other books that are mostly devoted to those subjects(measure theory and Lebesgue integration).
oh, I meant the 'material' it covers is relatively easy, not how it presents the material. May be this isn't true either, it's just the impression I got from my short time of studying.
Anyway thank you very much for your advice. I guess I will have to just study more to better see the bigger...
Thank you everybody who replied. I looked up the books mentioned above and was a bit puzzled.
Some books like Rudin's PMA or Apostol's cover the basic materials such as sequences or differentiation or Rieman-Stieltjes integration, while some other books I found cover very little of those, but...
It's not that I discovered a way to count it or anything, but I think I have some confusion about it.
I understand that Cantor set isn't countable and I accept the proof also.
But, what if we count the elements of the set like the following?
1, 0, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 1/27...
Hello,
I'm not quite sure if this kind of question can be posted on this board. Please excuse me if not.
I started studying real analysis with Rudin's Principles of Mathematics which was relatively compact. Then I bought Apostol's book which was much more helpful because it was more...
Thank you for the reply.
But what I'm trying to say is that the kind I mentioned above isn't the type that has the form of ∫f(g(x))*g'(x) dx, to which we can apply substitution rule directly.
And the solution kind of distorted the theorem and solved it in a weird way.
If ∫2(2x+4)^5 dx...
Say we are solving an indefinite integral ∫x√(2x+1) dx.
According to the textbook, the solution goes like this.
Let u = 2x+1. Then x = (u-1)/2.
Since √(2x+1) dx = (1/2)√u du,
x√(2x+1) dx = [(u-1)/2] * (1/2)√u du.
∫x√(2x+1) dx = ∫[(u-1)/2] * (1/2)√u du. <= What justifies this??
The...
Thank you very much!
There's one thing I'm still not sure about.
If it says f'(a) exists, can I take it that f is defined on some open interval that includes a?
Or does it cover the case where 'a' can be an end point of some close interval such as [a,b] on which f is defined?
Hi, I have a question about the definition of derivative.
As far as I know, for a real valued function f defined on a subset of R, the derivative of f at a is
(f(x+h)-f(x))/h as h → 0.
And if it exists it's said f is differentiable at x.
What if I define f : Q → R as follows...
Hi guys,
This is from so called ratio test, which says
The series Ʃa(n) diverges if abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer.
I was wondering if I could replace the condition with lim inf abs{a(n+1)/a(n)} ≥ 1.
So basically what I'm asking is whether these two below...