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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