Answer :
To determine whether the series [tex]\(\sum_{n=1}^{\infty} \frac{7^n + 9}{5^n}\)[/tex] converges or diverges, we will use the limit comparison test with the geometric series [tex]\(\sum_{n=1}^{\infty} \frac{7^n}{5^n}\)[/tex].
Here are the steps:
1. Consider the given series:
[tex]\[ \sum_{n=1}^{\infty} \frac{7^n + 9}{5^n} \][/tex]
2. Choose a comparison series:
We will compare this series with the geometric series
[tex]\[ \sum_{n=1}^{\infty} \frac{7^n}{5^n} \][/tex]
3. Define the terms of the series:
Let [tex]\(a_n = \frac{7^n + 9}{5^n}\)[/tex] for the given series and [tex]\(b_n = \frac{7^n}{5^n}\)[/tex] for the comparison series.
4. Compute the limit ratio:
To apply the limit comparison test, calculate:
[tex]\[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{7^n + 9}{5^n}}{\frac{7^n}{5^n}} = \lim_{n \to \infty} \frac{7^n + 9}{7^n} = \lim_{n \to \infty} \left( 1 + \frac{9}{7^n} \right) \][/tex]
5. Evaluate the limit:
Since [tex]\(\frac{9}{7^n}\)[/tex] approaches 0 as [tex]\(n\)[/tex] approaches infinity,
[tex]\[ \lim_{n \to \infty} \left( 1 + \frac{9}{7^n} \right) = 1 \][/tex]
6. Apply the limit comparison test:
The limit obtained is a positive, finite number [tex]\((=1)\)[/tex]. According to the limit comparison test, if [tex]\( \lim_{n \to \infty} \frac{a_n}{b_n} \)[/tex] is a positive finite number, then both series [tex]\(\sum a_n\)[/tex] and [tex]\(\sum b_n\)[/tex] either both converge or both diverge.
7. Determine the behavior of the comparison series:
The comparison series [tex]\(\sum_{n=1}^{\infty} \frac{7^n}{5^n}\)[/tex] is a geometric series with the common ratio [tex]\(\frac{7}{5}\)[/tex]. Since [tex]\(\left| \frac{7}{5} \right| > 1\)[/tex], the geometric series diverges.
8. Conclude the behavior of the given series:
Since the comparison series diverges and the limit comparison test provides a positive, finite limit, the given series
[tex]\[ \sum_{n=1}^{\infty} \frac{7^n + 9}{5^n} \][/tex]
also diverges.
Therefore, the correct answer is:
The series diverges.
Here are the steps:
1. Consider the given series:
[tex]\[ \sum_{n=1}^{\infty} \frac{7^n + 9}{5^n} \][/tex]
2. Choose a comparison series:
We will compare this series with the geometric series
[tex]\[ \sum_{n=1}^{\infty} \frac{7^n}{5^n} \][/tex]
3. Define the terms of the series:
Let [tex]\(a_n = \frac{7^n + 9}{5^n}\)[/tex] for the given series and [tex]\(b_n = \frac{7^n}{5^n}\)[/tex] for the comparison series.
4. Compute the limit ratio:
To apply the limit comparison test, calculate:
[tex]\[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{7^n + 9}{5^n}}{\frac{7^n}{5^n}} = \lim_{n \to \infty} \frac{7^n + 9}{7^n} = \lim_{n \to \infty} \left( 1 + \frac{9}{7^n} \right) \][/tex]
5. Evaluate the limit:
Since [tex]\(\frac{9}{7^n}\)[/tex] approaches 0 as [tex]\(n\)[/tex] approaches infinity,
[tex]\[ \lim_{n \to \infty} \left( 1 + \frac{9}{7^n} \right) = 1 \][/tex]
6. Apply the limit comparison test:
The limit obtained is a positive, finite number [tex]\((=1)\)[/tex]. According to the limit comparison test, if [tex]\( \lim_{n \to \infty} \frac{a_n}{b_n} \)[/tex] is a positive finite number, then both series [tex]\(\sum a_n\)[/tex] and [tex]\(\sum b_n\)[/tex] either both converge or both diverge.
7. Determine the behavior of the comparison series:
The comparison series [tex]\(\sum_{n=1}^{\infty} \frac{7^n}{5^n}\)[/tex] is a geometric series with the common ratio [tex]\(\frac{7}{5}\)[/tex]. Since [tex]\(\left| \frac{7}{5} \right| > 1\)[/tex], the geometric series diverges.
8. Conclude the behavior of the given series:
Since the comparison series diverges and the limit comparison test provides a positive, finite limit, the given series
[tex]\[ \sum_{n=1}^{\infty} \frac{7^n + 9}{5^n} \][/tex]
also diverges.
Therefore, the correct answer is:
The series diverges.
