# Tagged Questions

Convergence of sequences and different modes of convergence.

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### Convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$

Problem: Analyze the convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$. It seems to me that 'I' converges for $0<a<1$. My work: I wrote integral 'I' as a ...
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### Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
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### Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
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### Does this converge?

If I have $$X_i=\begin{cases}2\quad p=\frac{1}{3}\\ \frac{1}{2}\quad p=\frac{2}{3} \end{cases}$$ random variables with the same distribution. How can I compute the limit almost sure as $n\to\infty$ ...
### Does $\int_0^2\frac{1}{\ln(x)}dx$ converge?
Given the following integral: $$\int_0^2 \frac{1}{\ln(x)} dx$$ Does it converge? Iv'e gone this far:  \int_0^2 \frac{1}{\ln(x)} dx = \int_0^1 \frac{1}{\ln(x)} dx + \int_1^2 \frac{1}{\ln(x)} dx \$...