Question
Determine the convergence or divergence of the sequence with the given \( n \)th term. If the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.) \[ a_{n}=\frac{6}{n+9} \]
Ask by Stephens Garrett. in the United States
Dec 18,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
The sequence converges and its limit is 0.
Solution
To determine whether the sequence \( a_n = \frac{6}{n+9} \) converges or diverges, and to find its limit if it converges, we analyze its behavior as \( n \) approaches infinity.
1. **Examine the Denominator:**
- As \( n \) increases, \( n + 9 \) also increases without bound. Mathematically, \(\lim_{n \to \infty} (n + 9) = \infty\).
2. **Evaluate the Sequence:**
- Since the denominator grows without bound, the overall fraction \( \frac{6}{n+9} \) becomes smaller and smaller.
- Mathematically, \(\lim_{n \to \infty} \frac{6}{n+9} = 0\).
3. **Conclusion:**
- The sequence converges because it approaches a specific finite limit.
- The limit of the sequence is **0**.
**Answer:**
The sequence converges and its limit is 0
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Simplify this solution Beyond the Answer
To determine the convergence of the sequence \( a_{n}=\frac{6}{n+9} \), we need to analyze what happens to \( a_{n} \) as \( n \) approaches infinity. As \( n \) increases, the denominator \( n + 9 \) grows larger and larger. Consequently, the fraction as a whole approaches zero. Therefore, the limit of the sequence is: \[ \lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{6}{n+9} = 0. \] Thus, the sequence converges, and its limit is \( 0 \).
