# Tag Info

### Testing for convergence - $\int_{0}^{\infty} \frac{x^4}{e^{\sqrt{x}}} dx$

Hint The substitution $x = u^2, dx = 2 u \,du$, transforms the integral to $$2 \int_0^\infty u^9 e^{-u} \,du.$$
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### Convergence radius for Cauchy multiplication series

You were on the right track! Here's a way to successfully apply your idea . . . If $f,g$ are given by \begin{align*} f&= \frac{2-x}{(1-x)(3-x)} = \frac{1}{2(1-x)} + \frac{1}{2(3-x)} \\[4pt] g&...
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### Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and $A=\{x\in\mathbb{R} | * \}$. If $(x_n)\in A$ and converges to $c$, show $c\in A$.

The definition of continuity is that if $x_n\to c$, then $f(x_n)\to f(c)$. The only requirement is that $x_n$ and $c$ are in the domain of $f$ (otherwise the statement makes no sense because $f$ isn't ...
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### Does $\int_0^{+\infty}\frac{\ln(1+x^n)-\ln(1+x^{n+1})}{(1+x^2)\ln(x)}\, dx$ converge? How?

Mathematica suggests, that the integral does not depend on $n$ ...
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### Convergence of a Sequence by definition. Approach issue

You said in a comment that for certain values of $\varepsilon$ I am in no position to continue my argument since I never considered the sign of the right hand side expression in the first place I ...
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