Question

Suppose \[ a_n = \frac{3^n + 3}{5^n - 5} \quad \text{and} \quad b_n = \frac{1}{(1 + n^2)^{1/4}} \quad \text{for} \ n = 2, 3, 4, \ldots \] Then which one of the following is true?

• A. Both \(\sum_{n=2}^\infty a_n \) and \(\sum_{n=2}^\infty b_n \) are convergent.
• B. Both \(\sum_{n=2}^\infty a_n \) and \(\sum_{n=2}^\infty b_n \) are divergent.
• C. \(\sum_{n=2}^\infty a_n \) is convergent and \(\sum_{n=2}^\infty b_n \) is divergent.
• D. \(\sum_{n=2}^\infty a_n \) is divergent and \(\sum_{n=2}^\infty b_n \) is convergent.

Answer 1

The Correct Option is C

To determine the convergence or divergence of the series \( \sum_{n=2}^\infty a_n \) and \( \sum_{n=2}^\infty b_n \), we will analyze each series separately.

Analysis of \( \sum_{n=2}^\infty a_n \):

Given: \(a_n = \frac{3^n + 3}{5^n - 5}\).

As \( n \to \infty \), the dominant term in the numerator is \( 3^n \) and in the denominator is \( 5^n \). Therefore, we approximate:

\(a_n \approx \frac{3^n}{5^n} = \left(\frac{3}{5}\right)^n\).

This is a geometric series with common ratio \(r = \frac{3}{5}\), where \( |r| < 1 \). According to the properties of geometric series, a series converges if \( |r| < 1 \). Thus, \( \sum_{n=2}^\infty a_n \) is convergent.

Analysis of \( \sum_{n=2}^\infty b_n \):

Given: \(b_n = \frac{1}{(1 + n^2)^{1/4}}\).

As \( n \to \infty \), we approximate: \(b_n \approx \frac{1}{n^{1/2}}\).

We compare this with the p-series: \(\sum_{n=1}^\infty \frac{1}{n^p}\).

Where the series converges if \( p > 1 \) and diverges if \( p \leq 1 \). Here, \( p = \frac{1}{2} \leq 1 \), so the series \( \sum_{n=2}^\infty b_n \) diverges.

Conclusion:

Based on the analysis, \( \sum_{n=2}^\infty a_n \) is convergent and \( \sum_{n=2}^\infty b_n \) is divergent. Thus, the correct answer is: 

\(\sum_{n=2}^\infty a_n \) is convergent and \(\sum_{n=2}^\infty b_n \) is divergent.