Let an=∫01/nsin2xxdxa_n = \int_0^{1/n} \frac{\sin^2 x}{x} dxan=∫01/nxsin2xdx. Does the series ∑n=1∞an\sum_{n=1}^{\infty} a_n∑n=1∞an converge?
Yes, by the Comparison Test with ∑1n2\sum \frac{1}{n^2}∑n21.
Yes, by the Integral Test.
No, it diverges like the harmonic series.
No, it diverges by the Divergence Test.