does the integral $\displaystyle \int_0^1 {\frac{{e^x }}{{\sqrt {1 - \cos (x)} }}}$ converge or diverge???
2. Apply limit comparison test with $\displaystyle \int_0^1\frac{dx}{\sqrt x},$ which converges, hence $\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}}{\sqrt{1-\cos x}}\div \frac{1}{\sqrt{x}}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{x}{{e}^{x}}}{\sqrt{1-\cos x}}=\underset{x\to 0}{\mathop{\lim }}\,\sqrt{\frac{x}{1-\cos x}}\cdot {{e}^{x}}=0,$ since $\displaystyle \frac{x}{1-\cos x}\to0$ as $\displaystyle x\to0.$ Finally, the integral converges.