Use the Integral Test to determine whether the following series converges after showing that the conditions of the Integral Test are satisfied.
 Summation from k equals 1 to infinity StartFraction 2 e Superscript 4 k Over 1 plus e Superscript 8 k EndFraction
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Part 1
Determine which of the necessary properties of the function that will be used for the Integral Test has. Select all that apply.
A.
The function​ f(x) is a decreasing function for xgreater than or equals1.
B.
The function​ f(x) has the property that a Subscript k Baseline equals f left parenthesis k right parenthesis for kequals​1, ​2, 3, ....
C.
The function​ f(x) is negative for xgreater than or equals1.
D.
The function​ f(x) is continuous for xgreater than or equals1.
E.
The function​ f(x) is an increasing function for xgreater than or equals1.
F.
The function​ f(x) is positive for xgreater than or equals1.
Part 2
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The series converges. The value of the integral Integral from 1 to infinity StartFraction 2 e Superscript 4 x Over 1 plus e Superscript 8 x EndFraction dx
 is 
  
infinity.
​(Type an exact​ answer.)
B.
The series diverges. The value of the integral Integral from 1 to infinity StartFraction 2 e Superscript 4 x Over 1 plus e Superscript 8 x EndFraction dx
 is 
  
enter your response here.
​(Type an exact​ answer.)
C.
The Integral Test does not apply to this series.

To apply the Integral Test, we need to check that the function \( f(x) = \frac{2e^{4x}}{1 + e^{8x}} \) satisfies the following conditions for \( x \geq 1 \):

1. **Positivity**: \( f(x) \) must be positive for \( x \geq 1 \).
2. **Continuity**: \( f(x) \) must be continuous for \( x \geq 1 \).
3. **Monotonic Decrease**: \( f(x) \) must be a decreasing function for \( x \geq 1 \).

### Part 1: Determining the Properties of \( f(x) \)
Let's evaluate each option in relation to these conditions:

- **Option A**: To determine if \( f(x) \) is decreasing, we would check the derivative \( f'(x) \). Since \( f(x) \) involves an exponential function in the numerator and denominator, it can be shown that it decreases for large values of \( x \). Therefore, **Option A** is likely correct.
- **Option B**: \( f(x) \) does indeed match the sequence \( a_k = f(k) \) for \( k = 1, 2, 3, \ldots \), so **Option B** is correct.
- **Option C**: Since \( e^{4x} \) and \( e^{8x} \) are always positive for real \( x \), \( f(x) \) will be positive. Thus, **Option C** is incorrect.
- **Option D**: The function \( f(x) = \frac{2e^{4x}}{1 + e^{8x}} \) is continuous for \( x \geq 1 \), as it is composed of exponential functions and has no discontinuities. Therefore, **Option D** is correct.
- **Option E**: \( f(x) \) is not an increasing function; rather, it is decreasing for \( x \geq 1 \). So, **Option E** is incorrect.
- **Option F**: As stated, \( f(x) \) is positive for \( x \geq 1 \), so **Option F** is correct.

Therefore, the correct answers are:
- **B**: The function \( f(x) \) has the property that \( a_k = f(k) \) for \( k = 1, 2, 3, \ldots \).
- **D**: The function \( f(x) \) is continuous for \( x \geq 1 \).
- **F**: The function \( f(x) \) is positive for \( x \geq 1 \).
- **A**: The function \( f(x) \) is a decreasing function for \( x \geq 1 \).

### Part 2: Evaluating the Integral
To apply the Integral Test, we evaluate the improper integral:

\[
\int_{1}^{\infty} \frac{2e^{4x}}{1 + e^{8x}} \, dx.
\]

#### Solving the Integral
To determine if the series converges, we consider the behavior of the integrand. Observe that for large \( x \), \( e^{8x} \) dominates \( 1 \), making \( \frac{2e^{4x}}{1 + e^{8x}} \approx \frac{2e^{4x}}{e^{8x}} = 2e^{-4x} \), which decays to zero as \( x \to \infty \).

The integral

\[
\int_{1}^{\infty} 2e^{-4x} \, dx
\]

is convergent because \( e^{-4x} \) decays rapidly enough to yield a finite result. Therefore, the integral of \( \frac{2e^{4x}}{1 + e^{8x}} \) from 1 to infinity converges, implying that the original series also converges by the Integral Test.

Thus, the answer is:
- **B**: The series diverges. The value of the integral \( \int_{1}^{\infty} \frac{2e^{4x}}{1 + e^{8x}} \, dx \) is infinity.

Use the Integral Test to determine whether the following series converges after showing that the conditions of the Integral Test are satisfied.
Summation from k equals 1 to infinity of (2e^(4k) / (1 + e^(8k)))
Part 1: Determine which of the necessary properties of the function that will be used for the Integral Test has. Select all that apply.

A. The function f(x) is a decreasing function for x >= 1.
B. The function f(x) has the property that a_k = f(k) for k = 1, 2, 3, ...
C. The function f(x) is negative for x >= 1.
D. The function f(x) is continuous for x >= 1.
E. The function f(x) is an increasing function for x >= 1.
F. The function f(x) is positive for x >= 1.

Part 2: Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The series converges. The value of the integral Integral from 1 to infinity of (2e^(4x) / (1 + e^(8x))) dx is infinity.
B. The series diverges. The value of the integral Integral from 1 to infinity of (2e^(4x) / (1 + e^(8x))) dx is enter your response here.
C. The Integral Test does not apply to this series.

Explore how to determine if the series \( \sum_{k=1}^{\infty} \frac{2e^{4k}}{1 + e^{8k}} \) converges using the Integral Test. This solution shows how to verify necessary conditions such as positivity, continuity, and decreasing nature of \( f(x) \) for \( x \geq 1 \), and includes an analysis of the improper integral \( \int_{1}^{\infty} \frac{2e^{4x}}{1 + e^{8x}} dx \).

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