Intereting Posts

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the cone is contractible
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Is ${\rm conv}({\rm ext}((C(X))_1))$ dense in $(C(X))_1$?
An example showing that van der Waerden's theorem is not true for infinite arithmetic progressions
Proving a shifting inequality
Derivative on removable discontinuity
Uniform $L^p$ bound on finite measure implies uniform integrability
Maximizing the trace
Finding the largest subset of factors of a number whose product is the number itself
Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?
Example where union of increasing sigma algebras is not a sigma algebra
Can the cubic be solved this way?
Delta Dirac Function

Why sum of two little “o” notation is equal little “o” notation from sum?

$o( f(n) ) + o( g(n) ) = o( f(n) + g(n) ) ?$

For example:

- How prove this $\lim_{x\to+\infty}(f'(x)+f(x))=l$
- Improper rational/trig integral comes out to $\pi/e$
- Prove no existing a smooth function satisfying … related to Morse Theory
- What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$?
- Baby Rudin: Chapter 1, Problem 6{d}. How to complete this proof?
- Show that the set is closed

- $f(n) = n^3$
- $g(n) = 1/n$

so

- $o(f(n)) = n^2$
- $o(g(n)) = 1/n^2$

and

- $o( f(n) ) + o( g(n) ) = n^2 + 1/n^2$
- $o( f(n) + g(n) ) = n^2$

Of course, I could write it like

- $o( f(n) ) + o( g(n) ) = n^2 + o( g(n) )$
- $o( f(n) + g(n) ) = n^2 + o( g(n) )$

My question is why?

I don’t understand it, because in first we **always** get two parameters.

- Integral $\int_0^\pi \cot(x/2)\sin(nx)\,dx$
- Show that $\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$
- Limit of integration can't be the same as variable of integration?
- Sequence in $C$ with no convergent subsequence
- Sigma notation only for odd iterations
- What are the disadvantages of non-standard analysis?
- The closure of an open set in $\mathbb{R}^n$ is a manifold
- Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$
- Special arrows for notation of morphisms
- Does such a finitely additive function exist?

In this notation we always suppose that the function appeared in the parenthesis is positive, for a counter-example of this equality when this assumption is not applied, we can take

$1=o(n^2)$

and

$0=o(1-n^2)$ which contradicts with $1=o(1)$.

With this observations we have

$$

\left|\lim_{n\to \infty} \frac{o(f(n))+o(g(n))}{f(n)+g(n)}\right|\le \left|\lim_{n\to \infty} \frac{o(f(n))}{f(n)}+\lim_{n\to \infty} \frac{o(g(n))}{g(n)}\right|=0

$$

As we want. $\square$

- Prove that vector space and dual space have same dimension
- Help with the Neumann Problem
- Does a non-trivial solution exist for $f'(x)=f(f(x))$?
- Understanding quotients of $\mathbb{Q}$
- Diffeomorphisms and Lipschitz Condition
- Find three distinct triples (a, b, c) consisting of rational numbers that satisfy $a^2+b^2+c^2 =1$ and $a+b+c= \pm 1$.
- Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$
- Single variable complex analysis vs the world of the functions $f:\Bbb R^2 \to \Bbb R^2$.
- When does a polynomial divide $x^k – 1$ for some $k$?
- Simplify result of $\int_0^{\infty} \frac{1}{1+x^n}dx$
- Cracking Playfair code
- Equality of positive rational numbers.
- Potentially Useful Question
- Can a function be applied to itself?
- Evaluating limit $\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$