Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (If the quantity diverges, enter DIVERGES.) $a_n = 4 \sqrt{n} \sin\left(\frac{\pi}{\sqrt{n}}\right)$

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

Question :

Use a graph of the sequence to decide whether the sequence is convergent or divergent. if the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (if the quantity diverges, enter diverges.) $a_n = 4 \sqrt{n} \sin\left(\frac{\pi}{\sqrt{n}}\right)$

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

Use a graph of the sequence to decide whether the sequence is convergent or dive | Doubtlet.com

Solution:

Neetesh Kumar | December 31, 2024

Calculus Homework Help

This is the solution to Math 1c
Assignment: 11.1 Question Number 36
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Step-by-step solution:

Step 1: Simplify the sine term:

The argument of the sine function is $\frac{\pi}{\sqrt{n}}$ . As $n \to \infty$ , we know: $\frac{\pi}{\sqrt{n}} \to 0$

Using the Taylor expansion for $\sin(x)$ around $x = 0$ , we have: $\sin\left(\frac{\pi}{\sqrt{n}}\right) \approx \frac{\pi}{\sqrt{n}}$ for large $n$ .

Step 2: Substitute the approximation into $a_n$ :

Substitute $\sin\left(\frac{\pi}{\sqrt{n}}\right) \approx \frac{\pi}{\sqrt{n}}$ into the sequence: $a_n = 4 \sqrt{n} \cdot \frac{\pi}{\sqrt{n}}$

Simplify: $a_n = 4\pi$

Step 3: Conclude the behavior of the sequence:

As $n \to \infty$ , the sequence approaches a constant value of $4\pi$ . Therefore, the sequence converges.

Final Answer:

$\displaystyle\lim_{n \to \infty} a_n = \boxed{4\pi}$