State whether the sequence the set of terms with general term, the quotient of the sin of n and the square root of n converges or diverges, and if it converges, give its limit. (10 points)
A) Diverges
B) Converges to 1 and −1, depending on the value of the numerator
C) Converges to 1
D) Converges to 0
sin(n) / √n clearly approaches zero, since |sin(n)| <= 1 and √n→∞
I got diverges. Is this right? Please respond as soon as possible.
To determine whether the sequence formed by the general term, the quotient of sin(n) and the square root of n, converges or diverges, we need to analyze its behavior as n approaches infinity.
Let's denote the general term as a(n), which can be written as a(n) = sin(n) / √n.
To approach this problem, we can recall that the sine function (sin(x)) oscillates between -1 and 1 as the input (x) varies. The square root of n (√n) becomes larger as n increases.
Considering these observations, let's explore the options:
A) Diverges: If the sequence diverges, it means that the terms do not have a definite limit as n approaches infinity. However, since the sine function oscillates between -1 and 1, and the denominator √n increases as n increases, the quotient sin(n) / √n will not oscillate indefinitely. Hence, option A is not correct.
B) Converges to 1 and -1, depending on the value of the numerator: This option suggests that the limit of the sequence could be either 1 or -1, depending on the value of the numerator sin(n). However, since the sine function oscillates indefinitely and does not converge to a single value, the limit of the sequence cannot be 1 or -1. Therefore, option B is also incorrect.
C) Converges to 1: This option suggests that the limit of the sequence is 1. However, since the numerator sin(n) oscillates between -1 and 1, the sequence formed by the quotient sin(n) / √n will also oscillate indefinitely between a range of values. Consequently, this option is incorrect.
D) Converges to 0: This option suggests that the limit of the sequence is 0. Let's analyze this possibility further. As mentioned earlier, the numerator sin(n) oscillates between -1 and 1. Meanwhile, the denominator √n becomes larger as n increases. When we divide a number that oscillates between -1 and 1 by a number that increases without bound, the result will approach zero. Therefore, option D is correct.
Hence, the correct answer is D) Converges to 0.