Use the integral test to determine whether the series is convergent or divergent
[infinity]
σne−15n
n = 1
in order to apply the integral test for the function
f(x)
on the interval [a, [infinity]), it must be true that (choose all that apply)
a. f must be continuous.
b. f must be differentiable.
c. f must be positive.
d. f must be negative.
e. f must be increasing.
f. f must be decreasing.

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

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