$\int_{0}^{\infty}\frac{(x^{4}+2017)*(sin x)^{3}}{x^{5}+2018*x^{3}+1}dx$ How do I prove that it converges and that it does not converge absolutely?

I’m reading the proof of Riemann Rearrangement Theorem in T. Tao’s Analysis 1 textbook which can be found here Rearrangement Thm (the parts missing from the textbook, left as exercises for the reader, are completed by the user asking the question) but I don’t understand the last line of the proof where the user says […]

$$\sum_{n = 1}^{\infty}(-1)^{\frac{n(n+1)}{2}}(n^{\frac{1}{\sqrt[10]{n+10}}}-1)$$ Actually, I don’t have idea how to do it. Help me please Thanks a lot.

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_k = \lim_{n\to\infty} \sum_{k=1}^n a_{f(k)}, $$ where “$=$” is construed as meaning that if either limit exists then so does the other and in that case then they are equal? It is clear that there are uncountably many of these. Might […]

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