Question

Determine the behavior of the SEQUENCES. The sequence {n+sin(n²)/3 n}

Answer 1

The given sequence will oscillate indefinitely without approaching a specific number or infinity.

Step-by-step explanation:

The given sequence is {n+sin(n²)/3 n}. Let's break it down step by step:

1. Consider the expression in the sequence, n+sin(n²)/3 n.
2. Start by substituting n=1, n=2, n=3, etc., into the expression to generate the terms of the sequence.
3. Based on the behavior of sin(n²), the sequence will oscillate between values.
4. Since sin(n²) oscillates between -1 and 1, dividing it by 3 will make the oscillations smaller.
5. So, the sequence will exhibit a pattern of oscillation where the values of n become larger.
6. The sequence will not approach any specific number or go to infinity, but it will continue oscillating indefinitely.