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12h
answered Radius of convergence of power series of complex $\log$
1d
revised Definite integral, $\frac 1{\ln(x)}$
fix another typo
1d
revised Definite integral, $\frac 1{\ln(x)}$
fix signs
1d
revised Definite integral, $\frac 1{\ln(x)}$
fix typo
1d
revised Definite integral, $\frac 1{\ln(x)}$
mention the number of terms used in $(2)$
1d
revised Definite integral, $\frac 1{\ln(x)}$
add computation
2d
answered Definite integral, $\frac 1{\ln(x)}$
2d
comment A counter example
I have added a link to my answer.
2d
revised A counter example
add a link
2d
answered Proving the following number is real
2d
comment A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$
All the results one would use from Taylor series are in the four limits claimed at the beginning.
2d
comment A counter example
We don't use the monotonicity of the sequence $u_n$ to show that the limit of $u_n$ is $u$. That is obvious by the definition of $u_n$. We use the monotonicity of the sequence $u_n$ to guarantee that $\lim\limits_{n\to\infty}\|u_n\|_{L_{p^\ast}^1}=\infty$. Since $\|u_{n+1}\|_{L_{p^\ast}^1}\ge\|u_n\|_{L_{p^\ast}^1}$, if $\lim\limits_{n\to\infty}\|u_n\|_{L_{p^\ast}^1}\lt\infty$, then Monotone Convergence guarantees that $\|u\|_{L_{p^\ast}^1}\lt\infty$.
2d
revised A counter example
be more explicit
2d
comment A counter example
Since each $u_n$ is now continuous, we must have that $(2)$ is finite, but each one is monotonically increasing to $|x|^\alpha$, so the norm in $(2)$ grows without bound, yet the $C_\theta$ norm in $(1)$ stays at $1$.
2d
comment Find$\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 2x + 4}\,dx$ and $\int_{-\infty}^{\infty} \frac{\sin(x)}{x^2 + 2x + 4}\,dx$
@Mike: please unaccept my answer and, if you wish, accept Aaron Maroja's answer. It seems that I have included too much detail while trying to show that Jan Eerland's result-only answer has an error.
2d
comment Find$\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 2x + 4}\,dx$ and $\int_{-\infty}^{\infty} \frac{\sin(x)}{x^2 + 2x + 4}\,dx$
@AaronMaroja: If the OP will unaccept my answer, I will delete it. I've had things go funny when an accepted answer is deleted.
2d
revised A counter example
fix the norm symbol
2d
revised A counter example
fix answer to handle $u\in C(\overline{\Omega})$
2d
revised A counter example
fix answer to handle $u\in C(\overline{\Omega})$
2d
comment A counter example
Yes it is, as long as $\alpha+\theta\ge0$. Look at the definition of $C_\theta$.