There is a criterion to decide if the following integral converge or not?
$$\displaystyle\int_0^\infty \left(\dfrac12\right)^x\sin x\,dx$$
The idea is to avoid calculating its value.
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3$\int_{0}^{+\infty}2^{-x},dx = \frac{1}{\log 2}$ is convergent, so... – Jack D'Aurizio Oct 24 '18 at 19:23
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Thanks @JackD'Aurizio but $sin(x)$ is not a positive function, can I to use the comparison test in this case? – yemino Oct 24 '18 at 19:37
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2The value of the integral between $\int (-2^{-x}); dx$ and $\int 2^{-x}; dx$ both of which are finite. – irchans Oct 24 '18 at 19:46
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hint
$$|(\frac 12)^x\sin(x)|\le e^{-x\ln(2)}$$
By comparison test, the integral is absolutely convergent.
hamam_Abdallah
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