To determine where the given infinite series converges, we can use the concept of convergence of series. Specifically, we will use the Ratio Test.
The Ratio Test states that for a series Σ(aₙ) to converge, the limit of the absolute value of the ratio of consecutive terms, lim(n→∞) |aₙ₊₁ / aₙ|, must be less than 1.
In the given series, aₙ = 1/(9k^2 + 3k - 2). We need to find the limit of the absolute value of the ratio |aₙ₊₁ / aₙ| as n approaches infinity.
Let's compute the ratio:
|aₙ₊₁ / aₙ| = |(1/(9(k+1)^2 + 3(k+1) - 2)) / (1/(9k^2 + 3k - 2))|
= |(9k^2 + 21k + 9)/(9k^2 + 27k + 20)|
Simplifying the expression further:
|aₙ₊₁ / aₙ| = | [(3k + 1)(3k + 9)] / [(k + 4)(3k + 5)] |
= |(3k + 1)(3k + 9)| / |(k + 4)(3k + 5)|
Now, we want to find the limit of this expression as k approaches infinity.
As k approaches infinity, the highest power term dominates in each factor. We can ignore the lower-degree terms:
|aₙ₊₁ / aₙ| = |3k * 3k| / |k * 3k|
= | 9k^2 / 3k^2 |
= | 3 |
Taking the limit as k approaches infinity:
lim(k→∞) |aₙ₊₁ / aₙ| = lim(k→∞) 3 = 3
Since the limit is 3 (greater than 1), the Ratio Test tells us that the series diverges. Therefore, the given infinite series does not converge.