Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.) $a_n = \ln(n + 4) - \ln(n)$

Find:

$\displaystyle\lim_{n \to \infty} a_n = \boxed{?}$

Neetesh Kumar | December 29, 2024

Calculus Homework Help

This is the solution to Math 1c
Assignment: 11.1 Question Number 15
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Step-by-step solution:

Step 1: Simplify using logarithmic properties:

Using the logarithmic property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$, rewrite the sequence:

$a_n = \ln\left(\frac{n + 4}{n}\right)$

Simplify the fraction inside the logarithm:

$\frac{n + 4}{n} = 1 + \frac{4}{n}$

Thus:

$a_n = \ln\left(1 + \frac{4}{n}\right)$

Step 2: Evaluate the limit as $n \to \infty$:

As $n \to \infty$, $\frac{4}{n} \to 0$.

Substitute this into the simplified expression:

$\ln\left(1 + \frac{4}{n}\right) \to \ln(1)$

Since $\ln(1) = 0$, we have:

$\displaystyle\lim_{n \to \infty} a_n = 0$

Step 3: Conclusion:

Since the sequence approaches a finite value, it converges.

Final Answer:

1. The sequence $\boxed{\text{converges}}$
2. $\displaystyle\lim_{n \to \infty} a_n = \boxed{0}$

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