Answer:
A.
converges by integral test
Step-by-step explanation:
A. At first we need to verify that the function which the series is related (
) fills the necessary conditions to ensure that the test is effective.
*f(x) must be continuous or differentiable
*f(x) must be positive and decreasing
Let´s verify that
fills these conditions:
*Considering that eˣ≠0 for all x, the function
does not have any discontinuities, so it´s continuous
*Because eˣ is increasing:
if a<b ,then eᵃ<eᵇ
if 0<eᵃ<eᵇ ,then 1/eᵃ > 1/eᵇ
if 1/eᵃ > 1/eᵇ and a<b, then a/eᵃ<b/eᵇ
We conclude that
is decreasing
*Because eˣ is always positive and the sum is going from 1 to ∞, this show that
is positive in [1,∞).
Now we are able to use the integral test in
as follows:

Let´s proceed to integrate f(x) using integration by parts

Choose your U and dV like this:

And continue using the formula for integration by parts:




Because we are dealing with ∞, we´d rewrite it as a limit that will help us at the end of the integral:



We only have left to solve the limits, but because b goes to ∞ and it is in an exponential function on the denominator everything goes to 0


Showing that the integral converges, it´s the same as showing that the series converges.
By the integral test
converges