Answer
Function satisfies conditions for integral test
Series converges
Work Step by Step
In order integral test can be applied $\sum ^{\infty }_{n=1}a_{n}$
$f(x) $ must be positive ,continuous and decreasing for $x \geq 1$
We see that $f\left( x\right) =\dfrac {1}{2^{x}}$ is positive and continuous for $x \geq1$ and in order to find if the function is decreasing we need to find derivative of function
$f'\left( x\right) =-\dfrac {1}{2^{x}}\times \ln 2$
the function is decreasing
So it satisfies all three conditions for integral test
To find whether series is converges or diverges
$$ \int ^{\infty }_{1}\dfrac {1}{2^{x}}=\int ^{\infty }_{1}2^{-x}=\dfrac {-2^{-x}}{\ln 2}]^{\infty }_{1}=\dfrac {1}{2\ln 2} $$
So the series converges