Final Answer:
The sequence aₙ = (1 + 1/n)(-1)ⁿ consists of terms derived by alternately multiplying 1 + 1/n by (-1)ⁿ.
Step-by-step explanation:
In the given sequence, aₙ = (1 + 1/n)(-1)ⁿ, the term 1 + 1/n represents a fraction where the numerator is n + 1 and the denominator is n. Multiplying this fraction by (-1)ⁿ alternately changes the sign of each term in the sequence based on whether n is even or odd. When n is even, (-1)ⁿ is 1, leaving the term as 1 + 1/n. When n is odd, (-1)ⁿ is -1, resulting in the term -(1 + 1/n). This alternation creates a sequence with terms that oscillate in sign.
To gain more insight, let's compute the first few terms of the sequence:
a₁ = (1 + 1/1)(-1)¹ = 0,
a₂ = (1 + 1/2)(-1)² = -1/3,
a₃ = (1 + 1/3)(-1)³ = 2/9,
a₄ = (1 + 1/4)(-1)⁴ = -3/16.
The sequence aₙ exhibits an interesting behavior as n increases. The terms converge to 0, reflecting the impact of the (-1)ⁿ factor in alternating the sign of the terms. This alternating pattern results in the sequence oscillating around 0, with the magnitude of the terms diminishing as n approaches infinity. The sequence provides an example of how alternating signs in a sequence can affect its convergence behavior.