Question

Determine if the sequence aₙ=(−1)ⁿ⁻¹ (n-2)/(3n²+6) converges or diverges. If the sequence converges, to what value does it converge?

Answer 1

The sequence
`\(aₙ=(−1)ⁿ⁻¹ ((n-2))/((3n²+6))\)`converges to the value
`\(0\).`

Step-by-step explanation:

The given sequence is
`\(aₙ=(−1)ⁿ⁻¹ ((n-2))/((3n²+6))\).`To determine convergence or divergence, we analyze the behavior of
`\(aₙ\) as \(n\)`approaches infinity.

First, note that the term
`\((-1)ⁿ⁻¹\) alternates between \(-1\) and \(1\) as \(n\)`varies. The denominator
`\(3n²+6\) grows faster than the numerator \(n-2\), so as \(n\)` goes to infinity, the entire sequence approaches
`\(0\)`since the oscillations between
`\(-1\) and \(1\)` are outweighed by the diminishing fraction.

This behavior suggests convergence. To formalize this, we can use the limit comparison test, comparing
`\(aₙ\) with \((1)/(3n)\)`, as the leading term in the denominator. By the limit comparison test, both sequences behave identically, and since
`\((1)/(3n)\)` converges to
`\(0\), \(aₙ\)`also converges to
`\(0\).`

In conclusion, the sequence
`\(aₙ\)` converges, and its limit is
`\(0\)`.