6b ### Sequence and Series Analysis#### Example 1:Given the sequence:\[ a_n = \frac{5n^2 + 5n + 5}{5n^5 + 6n + 3} \]- Determine \( k = \) (Option to enter your answer)- Analyze the convergence or divergence of the series:\[ \sum_{n=1}^{\infty} a_n \]You have the option to select either "Converges" or "Diverges".#### Example 2:Given the sequence:\[ a_n = \left( \frac{5n^2 + 5n + 2}{5n^5 + 6n + 3\sqrt{n}} \right)^{10} \]- Determine \( k = \) (Option to enter your answer)- Analyze the convergence or divergence of the series:\[ \sum_{n=1}^{\infty} a_n \]You have the option to select either "Converges" or "Diverges".---### Analysis ProcessFor each given sequence:1. **Find the parameter \( k \)**: To identify \( k \), one must often explore the dominant behavior of the terms within the sequence, especially for large values of \( n \).2. **Convergence/Divergence Analysis**: To analyze whether the series converges or diverges, apply proper criteria such as Ratio Test, Root Test, Comparison Test, or Integral Test based on the structure of the sequence.Both examples aim to exercise your ability to manipulate and understand sequences and their corresponding infinite series, reinforcing concepts in real analysis and series convergence tests.

Name: 6b ### Sequence and Series Analysis#### Example 1:Given the sequence:\
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: 6b ### Sequence and Series Analysis#### Example 1:Given the sequence:\[ a_n = \frac{5n^2 + 5n + 5}{5n^5 + 6n + 3} \]- Determine \( k = \) (Option to enter your answer)- Analyze the convergence or divergence of the series:\[ \sum_{n=1}^{\infty} a_n \]