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 $f\left( x\right) =3^{-x}$ is positive and continuous for $x \geq1$ and in order to find if the function is decreasing we need to find derivative of function
$f'\left( x\right) =-3^{-x}\times \ln 3$
the function is decreasing
So it satisfies all three conditions for integral test
To find whether series converges or diverges, we see whether the integral converges or diverges.
$$\int ^{\infty }_{1}3^{-x}=\dfrac {-3^{-x}}{\ln 3}]^{\infty }_{1}=\dfrac {1}{3\ln 3} $$
So the series converges