Possible Duplicate:
Convergence/divergence of $\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$
Determine the improper integral $\int_0^{\infty} \frac{x-\sin x}{x^{7/2}}dx$ converge or diverge. Prove that please.
Possible Duplicate:
Convergence/divergence of $\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$
Determine the improper integral $\int_0^{\infty} \frac{x-\sin x}{x^{7/2}}dx$ converge or diverge. Prove that please.
Write for $0< \varepsilon<1$ $\int\limits_{0}^{+\infty}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}=\int\limits_{0}^{\varepsilon}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}+\int\limits_{\varepsilon}^{1}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}+\int\limits_{1}^{+\infty}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}.$ Since for small $x, \;\; 0
The second integral is proper integral, therefore it is finite.
The third integral converges, since $\left|\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\right| \leqslant \dfrac{x+1}{x^{\frac{7}{2}}} \leqslant \dfrac{2x}{x^{\frac{7}{2}}}=2x^{-\frac{5}{2}}.$