Determine whether the sequence converges or diverges. If it converges find the limit The provided image shows the general term of a sequence \( a_n \). The expression is given by:\[ a_n = \frac{(-1)^{n-1} n}{n^2 + 1} \]### Explanation:- **Numerator:** - \( (-1)^{n-1} \): This component introduces an alternating sign in the sequence. For even values of \( n \), this term equals 1, and for odd values of \( n \), this term equals -1. - \( n \): This is a linear term in the numerator, which affects the growth rate of the sequence term as \( n \) increases.- **Denominator:** - \( n^2 + 1 \): This quadratic term ensures that as \( n \) grows larger, the sequence term will decrease because the denominator increases more rapidly than the numerator.This sequence is particularly interesting because it combines oscillatory behavior (due to the alternating sign) with a decaying magnitude (due to the \( n^2 + 1 \) term in the denominator).### Usage:This sequence could be used in solving problems related to alternating series, convergence, and testing limits of sequences. It demonstrates how the interplay between the numerator and the denominator affects the overall behavior of the sequence. ### Mathematical SequenceThe following formula describes a mathematical sequence:\[ a_n = n \sin\left(\frac{1}{n}\right) \]In this sequence, \(a_n\) represents the \(n\)-th term, where \(n\) is a positive integer. The term is calculated by multiplying \(n\) by the sine of the reciprocal of \(n\).#### Explanation:- \( n \): Represents the position in the sequence (i.e., 1st term, 2nd term, etc.).- \( \sin \): The sine function, which is a trigonometric function.- \( \frac{1}{n} \): The reciprocal of \(n\).#### Analysis:- As \(n\) increases, \(\frac{1}{n}\) becomes very small.- The sine of a very small number approaches that number, such that \(\sin\left(\frac{1}{n}\right) \approx \frac{1}{n}\) for large \(n\).- Therefore, for large \(n\), the term \( a_n \) approximately equals \(n \cdot \frac{1}{n} = 1\).This indicates that \(a_n\) approaches 1 as \(n\) becomes large. This type of behavior is useful in mathematical analysis, particularly in the context of sequences and series to understand their convergence properties.

Name: Determine whether the sequence converges or diverges. If it converges
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: Determine whether the sequence converges or diverges. If it converges find the limit The provided image shows the general term of a sequence \( a_n \). The expression is given by:\[ a_n = \frac{(-1)^{n-1} n}{n^2 + 1} \]### Explanation:- **Numerator