Questions: Problem 1. (3 points) Consider the sequence an where the n^th term is given by an=(3 n+5)/(4 n+7) - Part 1: Limit of the Sequence Find the limit of this sequence: lim n -> infinity (3 n+5)/(4 n+7)=□ Σ Remember: INF, -INF, DNE are also possible answers. - Part 2: Converge or Diverge? Does this sequence converge or diverge? ? □

Solution Steps

To solve the given problem, we need to analyze the behavior of the sequence as \( n \) approaches infinity.

To find the limit of the sequence \( a_n = \frac{3n + 5}{4n + 7} \) as \( n \) approaches infinity, we can divide the numerator and the denominator by \( n \). This simplifies the expression and allows us to see the dominant terms.

A sequence converges if it approaches a finite limit as \( n \) approaches infinity. If the limit found in Part 1 is a finite number, the sequence converges; otherwise, it diverges.

To find the limit of the sequence \( a_n = \frac{3n + 5}{4n + 7} \) as \( n \) approaches infinity, we simplify the expression by dividing both the numerator and the denominator by \( n \):

\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{3 + \frac{5}{n}}{4 + \frac{7}{n}} \]

As \( n \) approaches infinity, the terms \( \frac{5}{n} \) and \( \frac{7}{n} \) approach \( 0 \). Thus, we have:

\[ \lim_{n \to \infty} a_n = \frac{3 + 0}{4 + 0} = \frac{3}{4} \]

A sequence converges if it approaches a finite limit as \( n \) approaches infinity. Since we found that:

\[ \lim_{n \to \infty} a_n = \frac{3}{4} \]

which is a finite number, we conclude that the sequence converges.

Final Answer