Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.) $a_n = \frac{\sin(5n)}{5 + \sqrt{n}}$

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

Neetesh Kumar | December 31, 2024

Calculus Homework Help

This is the solution to Math 1c
Assignment: 11.1 Question Number 35
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Step-by-step solution:

Step 1: Analyze the numerator and denominator:

1. The numerator $\sin(5n)$ oscillates between $-1$ and $1$ for all $n$.
2. The denominator $5 + \sqrt{n}$ grows without bound as $n \to \infty$, since $\sqrt{n} \to \infty$.

Step 2: Consider the absolute value of $a_n$:

The absolute value of $a_n$ is:

$|a_n| = \left|\frac{\sin(5n)}{5 + \sqrt{n}}\right|$

Since $|\sin(5n)| \leq 1$, we have:

$|a_n| \leq \frac{1}{5 + \sqrt{n}}$

As $n \to \infty$, the denominator $5 + \sqrt{n} \to \infty$, which implies:

$|a_n| \to 0$

Step 3: Apply the squeeze theorem:

From the inequality $|a_n| \leq \frac{1}{5 + \sqrt{n}}$, and since $\frac{1}{5 + \sqrt{n}} \to 0$ as $n \to \infty$, it follows that: $a_n \to 0$

Step 4: Conclude the behavior of the sequence:

The sequence converges, and its limit is:

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

Final Answer:

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

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