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