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 Murray Reyes. in the United States

Dec 18,2024

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.) $$ a_{n}=5^{1 / n} $$

UpStudy AI · Accepted Answer

To determine whether the sequence $$ a_n = 5^{1/n} $$ converges or diverges, let's analyze its behavior as $$ n $$ approaches infinity.

### Step-by-Step Analysis:

1. **Understanding the Sequence:**
   - The sequence is defined as $$ a_n = 5^{1/n} $$.
   - This can be rewritten using the exponential function: $$ a_n = e^{\ln(5)/n} $$.

2. **Taking the Limit:**
   - To find the limit as $$ n \to \infty $$, consider:
     $$
     \lim_{n \to \infty} 5^{1/n} = \lim_{n \to \infty} e^{\ln(5)/n}
     $$
   - As $$ n $$ becomes very large, $$ \ln(5)/n $$ approaches 0:
     $$
     \lim_{n \to \infty} e^{\ln(5)/n} = e^0 = 1
     $$

3. **Conclusion:**
   - The sequence $$ a_n = 5^{1/n} $$ approaches 1 as $$ n $$ becomes large.

### Final Answer:
$$
\boxed{1}
$$
The sequence $$ a_n = 5^{1/n} $$ converges to 1 as $$ n $$ approaches infinity.

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.)