query: $\int_{0}^{+\infty} \frac{dx}{f(x)}$ converges if and only if $\int_{0}^{+\infty} \frac{x}{F(x)}dx$ converges
positive_passages.docid: 142854
positive_passages.text: Convergence of Integrals - A Problem
negative_passages.docid: 87169
negative_passages.text: Convergence of $\int_{1}^{\infty}x\cdot \sin{(e^x})\,dx$
negative_passages.docid: 151149
negative_passages.text: $\lim_{n \to \infty} n \int_{1}^{\infty} \frac{dx}{1+x^n}$, solution critique needed
negative_passages.docid: 301
negative_passages.text: Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.
negative_passages.docid: 63756
negative_passages.text: Convergence or divergence of the integral $\int\limits_{10}^\infty \frac{e^x}{x^{1 + \frac{1}{x}} (e^x -1)} \,dx$
negative_passages.docid: 58361
negative_passages.text: Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?
negative_passages.docid: 12480
negative_passages.text: Convergence of $\sum_{n=1}^{\infty} \frac{\frac{1}{2}+(-1)^n}{n}$
negative_passages.docid: 3744
negative_passages.text: If $\sum_{n=1}^{\infty} a_n$ converges, then so does $\sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$
negative_passages.docid: 102012
negative_passages.text: Determine if $\sum_{n=2}^{\infty} \frac{(-1)^n}{n+(-1)^n}$ converges or diverges
negative_passages.docid: 114702
negative_passages.text: $\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$?
negative_passages.docid: 62892
negative_passages.text: Uniform convergence of $\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$
negative_passages.docid: 135017
negative_passages.text: Test $f_n = (1+\frac{1}{n^2})^n$ for convergence and give its limit if it exists.
negative_passages.docid: 97039
negative_passages.text: Use integral test to show that Gaussian Integral $\int_{-\infty}^{\infty} e^{-x^2} \ dx$ converges?
negative_passages.docid: 117196
negative_passages.text: test for convergence $\sum_{n=1}^{\infty} \frac{1}{n^n}$
negative_passages.docid: 17028
negative_passages.text: How to test if $\sum_{n=1}^{\infty }\frac{1}{n^{1+\frac{1}{n}}}$ converges?
negative_passages.docid: 38062
negative_passages.text: Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$
negative_passages.docid: 56867
negative_passages.text: Convergence proof that involves integral