To determine whether the series

$\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \frac{2}{4n + 5e^n}$

converges or diverges using the Limit Comparison Test, we will proceed as follows:

1. Choose a Comparison Series:
   Since the denominator $4n + 5e^n$ grows very quickly due to the exponential term $e^n$, the term $\frac{2}{5e^n}$ will eventually dominate for large $n$. Thus, we can compare $a_n$ to a similar term:

   $b_n = \frac{2}{5e^n} = \frac{2}{5} \cdot e^{-n}.$

   This series resembles a geometric series, $\sum ar^{n-1}$, where $a = \frac{2}{5}$ and $r = \frac{1}{e}$.

2. Apply the Limit Comparison Test:
   To use the test, we find

   $\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{2}{4n + 5e^n}}{\frac{2}{5e^n}} = \lim_{n \to \infty} \frac{5e^n}{4n + 5e^n}.$

   Dividing the numerator and denominator by $e^n$, we get:

   $\lim_{n \to \infty} \frac{5}{\frac{4n}{e^n} + 5}.$

   Since $\frac{4n}{e^n} \to 0$ as $n \to \infty$, the limit becomes:

   $\lim_{n \to \infty} \frac{5}{0 + 5} = 1.$

   Since the limit is a positive finite number (1), the Limit Comparison Test applies.

3. Determine Convergence of $b_n$:
   The series $\sum b_n = \sum \frac{2}{5} e^{-n}$ is a geometric series with ratio $r = \frac{1}{e} < 1$. Therefore, $\sum b_n$ converges.

4. Conclusion:
   By the Limit Comparison Test, since $\sum b_n$ converges, the series $\sum a_n$ also converges.

Final Answers:

• Comparison series $b_n$: $\frac{2}{5} e^{-n}$ with $a = \frac{2}{5}$ and $r = \frac{1}{e}$.
• Limit: $\lim_{n \to \infty} \frac{a_n}{b_n} = 1$.
• Convergence: The series $\sum a_n$ converges by the Limit Comparison Test.

Would you like more details on any part of this explanation?

Follow-up Questions

1. What is the general form of the Limit Comparison Test?
2. How does the exponential function $e^n$ influence the growth rate of sequences?
3. What properties define a geometric series, and why is convergence determined by $|r| < 1$?
4. Could a different comparison series be used here, and would it change the conclusion?
5. What other convergence tests could be applied to a series with exponential terms?

Tip

When choosing a comparison series, look for terms in $a_n$ that dominate as $n \to \infty$. This will often lead you to an effective comparison series.