ok $\int_0^{1}\frac{cos(x)}{x^2}dx=\int_0^{1}\sum_{n=1 }^{\infty}\frac{(-1)^{n}x^{2n-2}}{(2n)!}dx=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n-1}}{(2n-1)\cdot{(2n)!}}$ which converges from 0 to 1
Yes it is $\frac{cos(x)}{x^2}=\frac{\sum_{n=0}^{\infty}\frac{ (-1)^{n}x^{2n}}{(2n)!}}{x^2}=\sum_{n=1}^{\infty}\fra c{(-1)^{n}x^{2n-2}}{(2n)!}$