How can I prove that $\int_{0}^{\infty}\frac{\sin(x)}{x^{\lambda}}$ converges for $\lambda \in (0,2)$?
117264
Prove that $\int_0^{\infty} \frac{\sin x}{x^p}\, dx$ converges for $0<p<2$
124916
$f$ such that $\int_1^{\infty}f(x)dx$ converges, but not absolutely?
121979
Pointwise convergence of sine series of $x^{-2}$
38062
Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$
44288
Is my proof of convergence for $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ true?
90558
A question related to a convergence of an integral
41751
$\sum_{n=1}^\infty \frac{i^n}{n}$ converge?
94910
How does one show that $\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x$ converges?
100744
Proof for divergence of $\int_1^\infty \cos(x^\frac{3}{4})$
153267
Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.
56867
Convergence proof that involves integral
80465
Convergence/Divergence of $\sum_{n=1}^{\infty} \sin(1/n)$
16480
Does $\sum_{n=1}^{\infty}\ln(n\sin(\frac{1}{n}))$ converge?
21970
Convergence of the Power Series for $xu''+\sin(x)u=0$
114702
$\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$?
88792
Convergence of $\int_0^\infty $sin$ (x^p) dx$
10547
$\int_0^\infty \frac{\sin(x^2+ax)}{x}\,dx$ converges