Finding whether the series $$\int^{\infty}_{1}e^{x}\cos (x)\cdot x^{-\frac{1}{2}}dx$$ converges or diverges.
What i try::
$$I=\int^{\infty}_{1}e^{x}\cos(x)\cdot x^{-\frac{1}{2}}dx\leq \int^{\infty}_{1}e^{x}\cdot x^{-\frac{1}{2}}dx$$
Now put $x=t^2$ and $dx=2tdt$
$$I<2\int^{\infty}_{1}e^{t^2}dt<\infty$$
From series expansion.
But this does not show anything.
Hiw do i solve it. Help me please. Thanks.
Due to the fact that $e^x \cos x$ diverges away from $0$ arbitrarily often, and $\frac{e^x}{\sqrt{x}}$ is an increasing function (since $e^x = \omega(\sqrt{x})$, the integrand doesn't $\to 0$ as $x \to \infty$. Hence, integral diverges.