Intereting Posts

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Computing the homology groups.
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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?

- Why does $L^2$ convergence not imply almost sure convergence
- Geometric interpretation of Euler's identity for homogeneous functions
- Definition of Equivalent Norms
- Inverse of the sum $\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$
- $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$
- Is the graph $G_f=\{(x,f(x)) \in X \times Y\ : x \in X \}$ a closed subset of $X \times Y$?

Thanks!

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

- for infinite compact set $X$ the closed unit ball of $C(X)$ will not be compact
- Is it meaningful to take the derivative of a function a non-integer number of times?
- Is $f(x)=\sup_{y\in K}g(x, y)$ a continuous function?
- Determine whether $\sum_{n=1}^\infty \frac {(-1)^n|\sin(n)|}{n}$ converges
- Prove that if $f:A\to B$ is uniformly continuous on $A$ and $g$ is uniformly continuous on $B$, then $g(f(x))$ is uniformly continuous on $A$
- Minkowski Dimension of Special Cantor Set
- Converse of interchanging order for derivatives
- Proving that the given two integrals are equal
- Convergence of the improper integral $\int_{0}^{\pi/2}\tan^{p}(x) \; dx$
- If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?

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

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