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Feb
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Jan
23
comment Integrating a Taylor series term-by-term
Well you actually have 3 limits going on, so maybe its the limit of $z \to \infty$ that is resisting the swap. $$\int_0^\infty \frac{\sin x}{x} dx = \lim_{z\to\infty}\int_{1/z}^z \frac{\sin x}{x} dx = \lim_{z \to \infty}\lim_{n \to \infty}\sum_{k = 0}^n\frac{\sin x_k^*}{x_k^*} \Delta x_k^*.$$ Now using the Taylor series, we have, $$\lim_{z \to \infty}\lim_{n \to \infty}\lim_{m \to \infty}\sum_{k = 0}^n \sum_{j=0}^m \frac{(-1)^j}{(2j + 1)!} (x_k^*)^{2j} \Delta x_k^*.$$ Note: $x_k^*$ depends on $z$.
Jan
23
comment Integrating a Taylor series term-by-term
When converting from the left hand side to the right hand side, you are swapping limits. I am not sure if that is exactly the problem here, but you certainly have to be careful with it. For example, $$\lim_{m\to\infty}\lim_{n\to\infty} \frac{n}{m + n} = \lim_{m \to \infty} 1 = 1,$$ but $$ \lim_{n\to\infty}\lim_{m\to\infty} \frac{n}{m + n} = \lim_{n \to \infty} 0 = 0.$$
Jan
23
answered Graph Theory: Simple Graph
Dec
26
revised Open Sets Definition
added 7 characters in body
Dec
26
answered Open Sets Definition
Dec
11
answered Prime Factors problem and combination math.
Dec
11
comment What does $\frac{1}{n}$ converge to?
I went to a local community college, so I am surprised that it would be much fancier than any other school. The standard calculus books Thomas and Stewart both have early chapters on $\epsilon-\delta$ proofs. So I guess people just skip it or something. (They also have proofs throughout the book, but I suppose one could just use these books as a problem book and ignore the fact they have "reading parts", heh)
Dec
11
comment What does $\frac{1}{n}$ converge to?
@DonAnselmo, I learned calculus in the US and had proofs. I think that post is overgeneralizing.
Dec
11
comment What does $\frac{1}{n}$ converge to?
Do people cover convergence before calculus now? How do they define what it means? EDIT: Based on other posts by the OP, I assume they are in a calculus course, so $\epsilon-\delta$ should be fair game there.
Dec
11
answered What does $\frac{1}{n}$ converge to?
Dec
9
revised Solving $ F(x,y) = \int_0^x\int_0^y (u+v)dudv ?$
Fixed title to not be multilined
Dec
9
suggested suggested edit on Solving $ F(x,y) = \int_0^x\int_0^y (u+v)dudv ?$
Dec
9
answered Solving $ F(x,y) = \int_0^x\int_0^y (u+v)dudv ?$
Dec
9
revised Square-integrable functions, proofs of $L^2-norm$ properties
deleted 44 characters in body
Dec
9
answered Square-integrable functions, proofs of $L^2-norm$ properties
Dec
9
answered Additional solutions to quadratic equations which don't match the formula answer.
Dec
9
answered Using Cantor's Diagonalization to Show Polynomials are NOT Countable
Dec
9
answered How do I prove this property about symmetric matrices?
Dec
9
comment Expanding brackets of the form $(a+b)^n$
@LinearAlgebra I do think that is an easier proof to check for the correctness of the formula, but it kind of makes the formula feel like magic because it provides no intuition on how one might derive that formula. The way the OP is learning it is a bit more instructive on that part.