%5En-(3n-1)%2F(2n%2B1).jpeg "(-1)^n (3n-1)/(2n+1)")
Exploring the Sequence (-1)^n (3n-1)/(2n+1)
This article delves into the fascinating properties of the sequence defined by the formula (-1)^n (3n-1)/(2n+1). We will analyze its behavior, convergence, and some interesting characteristics.
Understanding the Sequence
The sequence is defined for all natural numbers n (n = 1, 2, 3, ...). It exhibits an alternating pattern due to the (-1)^n term. This term causes the signs of the sequence to alternate between positive and negative for successive values of n.
Let's analyze the first few terms of the sequence:
- n = 1: (-1)^1 (31-1)/(21+1) = -2/3
- n = 2: (-1)^2 (32-1)/(22+1) = 5/5 = 1
- n = 3: (-1)^3 (33-1)/(23+1) = -8/7
- n = 4: (-1)^4 (34-1)/(24+1) = 11/9
We can observe that the sequence alternates in sign and the absolute value of the terms appears to be increasing.
Investigating Convergence
To determine whether the sequence converges or diverges, we can examine the limit as n approaches infinity.
lim (n -> ∞) (-1)^n (3n-1)/(2n+1)
We can simplify this expression by dividing both numerator and denominator by n:
lim (n -> ∞) (-1)^n (3 - 1/n)/(2 + 1/n)
As n approaches infinity, 1/n approaches 0. Therefore, the limit becomes:
lim (n -> ∞) (-1)^n (3 - 0)/(2 + 0) = lim (n -> ∞) (-1)^n (3/2)
The term (-1)^n oscillates between -1 and 1 as n approaches infinity. This means the sequence does not approach a single value and therefore diverges.
Interesting Observations
- Alternating signs: The sequence exhibits alternating signs due to the (-1)^n factor. This alternating nature is a key characteristic of the sequence.
- Increasing absolute value: The absolute value of the terms generally increases as n increases. This can be seen by analyzing the expression (3n-1)/(2n+1). As n gets larger, the numerator grows faster than the denominator, leading to larger absolute values.
- No specific limit: The sequence diverges as it does not approach a single value as n approaches infinity. This divergence is caused by the oscillating nature of the (-1)^n term.
Conclusion
The sequence (-1)^n (3n-1)/(2n+1) is a fascinating example of a sequence that diverges due to its alternating nature. Analyzing its terms and investigating its convergence provides valuable insights into the behavior of sequences.