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Let's analyze the given series:

\[
\sum_{n=1}^{\infty} \frac{5e^n}{1 + e^{2n}}
\]

### Step 1: Simplify the General Term

The general term of the series is:

\[
a_n = \frac{5e^n}{1 + e^{2n}}
\]

For large values of \(n\), notice that \(e^{2n}\) dominates over 1, so we can approximate the denominator as \(e^{2n}\). This gives:

\[
a_n \approx \frac{5e^n}{e^{2n}} = \frac{5}{e^n}
\]

Thus, for large \(n\), the term \(a_n\) behaves like a geometric series with ratio \(r = \frac{1}{e}\), which is less than 1 (since \(e > 1\)).

### Step 2: Conclusion on Convergence

Since the general term of the series behaves like \(\frac{5}{e^n}\) for large

Does the series below converge or diverge? Give a reason for your answer.

∑ (5e^n) / (1 + e^(2n)) from n = 1 to infinity.

Choose the correct choice below:
A. The Integral Test shows that the series diverges.
B. The series converges because it is a geometric series with |r| < 1.
C. The series diverges because it is a geometric series with |r| ≥ 1.
D. The Integral Test shows that the series converges.

Determine whether the series (5e^n) / (1 + e^(2n)) converges or diverges. Learn how to analyze the behavior of the series using geometric series tests and approximation methods, suitable for college-level calculus.

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