Intereting Posts

Find limit $a_{n + 1} = \int_{0}^{a_n}(1 + \frac{1}{4} \cos^{2n + 1} t)dt,$
Finding the intersection point of many lines in 3D (point closest to all lines)
Solving an ODE from a PDE
If $\omega$-chains corresponds to maximality, then $\kappa$-chains corresponds to ???
How to calculate the PSD of a stochastic process
Irreducibility of $X^{p-1} + \cdots + X+1$
If $dF_p$ is nonsingular, then $F(p)\in$ Int$N$
Parameters on $SU(4)$ and $SU(2)$
Why is $\sqrt{2\sqrt{2\sqrt{2\cdots}}} = 2$?
$P(x)\in\mathbb Z$ iff $Q(x)\in\mathbb Z$
Question about weak convergence, $\lbrace f(x_{n}) \rbrace$ converges for all $n$, then $x_{n} \rightharpoonup x$
Distribution of $(XY)^Z$ for $(X,Y,Z)$ i.i.d. uniform on $(0,1)$
Proof that there is no continuous 1-1 map from the unit circle in $\Bbb R^2$ to $\Bbb R$.
Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only
Number of binary numbers with two consecutive zeros

Lets Think of this: I posted a question and got a wonderful answer by a smart user, but I couldn’t understand part of the method.

$$\begin{align}

\sum_{n=1}^{\infty}\frac{1}{9n^2 + 3n – 2}

&=\frac{1}{3}\sum_{n=1}^{\infty}\left(\frac{1}{3n – 1}-\frac{1}{3n + 2}\right)\\\\

&=\frac{1}{3}\sum_{n=1}^{\infty}\int_0^1\left(x^{3n-2}-x^{3n+1}\right){\rm d}x\\\\

&=\frac{1}{3}\int_0^1\sum_{n=1}^{\infty}\left(x^{3n-2}-x^{3n+1}\right){\rm d}x\\\\\end{align}$$

How is it possible to interchange the summation and integral?

- A sum for stirling numbers Pi, e.
- Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?
- Characterization of Dirac Masses on $C(,\mathbb{R}^d)$
- Integral eigenvectors and eigenvalues
- Question about Feller's book on the Central Limit Theorem
- A “clean” approach to integrals.

Thanks!

And in general, for what does this apply? Thank you very much!

- T-invariant sub-sigma algebra
- Fractional Trigonometric Integrands
- Moving a limit inside an infinite sum
- How to solve an integral with a Gaussian Mixture denominator?
- How find this equation $\prod\left(x+\frac{1}{2x}-1\right)=\prod\left(1-\frac{zx}{y}\right)$
- Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$
- Find the latus rectum of the Parabola
- Calculate:$y'$ for $y = x^{x^{x^{x^{x^{.^{.^{.^{\infty}}}}}}}}$ and $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+…\infty}}}}$
- Showing a compact metric space has a countable dense subset
- Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$

You’re allowed to do this anytime the series is uniformly convergent.

In uniform convergence, you tell me the $\epsilon$ that you want the *entire* partial sum to be within for the entire graph, and I give you an $M$ that guarantees you can get within $\epsilon$ if you choose $n > M$, but the $M$ must work anywhere on the graph, and not be dependent on which point you choose. In regular (called pointwise) convergence, the $\epsilon-M$ guarantee can use a different $M$ at different points on the graph.

Pointwise vs. Uniform Convergence

- How to prove that compact subspaces of the Sorgenfrey line are countable?
- Textbook for Projective Geometry
- Conditions for Taylor formula
- How to solve implicit differential equation?
- Equal slicing of my spherical cake
- Partial limits of sequence
- Prove that $T$ is an orthogonal projection
- Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$
- Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$
- Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?
- $f$ is a real function and it is $\alpha$-Holder continuous with $\alpha>1$. Is $f$ constant?
- Explicit isomorphism $S_4/V_4$ and $S_3$
- How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
- Operator norm of the inverse
- equation of a curve given 3 points and additional (constant) requirements