Determine the convergence or divergence of the sequence with the given th term. If the sequence converges, find its limit. (If
the quantity diverges, enter DIVERGES.)

Ask by Schofield Luna. in the United States

Dec 18,2024

Question

Determine the convergence or divergence of the sequence with the given th term. If the sequence converges, find its limit. (If
the quantity diverges, enter DIVERGES.)

Ask by Schofield Luna. in the United States

Dec 18,2024

UpStudy AI · Accepted Answer

To determine whether the sequence $$ a_{n} = \frac{\ln(n^{9})}{4n} $$ converges or diverges, and to find its limit if it converges, follow these steps:

1. **Simplify the Sequence:**
   $$
   a_{n} = \frac{\ln(n^{9})}{4n} = \frac{9 \ln(n)}{4n}
   $$
   So, the sequence simplifies to:
   $$
   a_{n} = \frac{9}{4} \cdot \frac{\ln(n)}{n}
   $$

2. **Evaluate the Limit:**
   We need to find:
   $$
   \lim_{n \to \infty} \frac{\ln(n)}{n}
   $$
   As $$ n $$ approaches infinity, $$ \ln(n) $$ grows much slower than $$ n $$. Therefore:
   $$
   \lim_{n \to \infty} \frac{\ln(n)}{n} = 0
   $$

3. **Determine the Limit of the Sequence:**
   $$
   \lim_{n \to \infty} a_{n} = \frac{9}{4} \cdot 0 = 0
   $$

**Conclusion:**  
The sequence $$ a_{n} = \frac{\ln(n^{9})}{4n} $$ **converges** and its limit is **0**.

**Answer:**  
The sequence converges and its limit is 0
The sequence $$ a_{n} = \frac{\ln(n^{9})}{4n} $$ converges and its limit is 0.

Determine the convergence or divergence of the sequence with the given th term. If the sequence converges, find its limit. (If
the quantity diverges, enter DIVERGES.)

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Determine the convergence or divergence of the sequence with the given th term. If the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.)

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Determine the convergence or divergence of the sequence with the given th term. If the sequence converges, find its limit. (If
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