Test the convergence of $\int_{0}^{1}\frac{\sin(1/x)}{\sqrt{x}}dx$
What I did
- Expanded sin (1/x) as per Maclaurin Series
- Divided by $\sqrt{x}$
- Integrate
- Putting the limits of 1 and h, where h tends to zero
So after step 3, I get something like this:
$S= \frac{-2}{\sqrt{x}}+\frac{2}{5\cdot 3! x^{5/2}}- \frac{2}{9 \cdot 5!x^{9/2}}+\frac{2}{13\cdot 7!x^{13/2}}-...$ Putting Limits: $I=S(1)-S(0)$ But I am stuck at calculating $S(0)$