Bounding the Sequence a_n (2n - 7) / (3n^2)

In the realm of mathematical sequences, determining whether a given sequence is bounded is a fundamental concept. Specifically, for the sequence defined by an (2n - 7) / (3n^2), we will explore its behavior as n approaches infinity and check if it remains within certain limits for all n.

To determine if the sequence is bounded, we need to analyze its behavior as n approaches infinity and also check if it stays within certain limits for all n. Let's break down the process step by step.

Step 1: Analyze the Limit as n Approaches Infinity

The first step is to find the limit of the sequence as n approaches infinity. This helps us understand the long-term behavior of the sequence. The limit is given by:

[lim_{n to infty} a_n lim_{n to infty} frac{2n - 7}{3n^2}]

To evaluate this limit, we can divide the numerator and the denominator by n^2:

[lim_{n to infty} frac{frac{2n}{n^2} - frac{7}{n^2}}{3 frac{2}{n}} frac{0 - 0}{3 0} frac{0}{3} frac{2}{3}]

Step 2: Boundedness of the Sequence

A sequence is bounded if there exists a real number M such that an ≤ M for all n. To check the boundedness, we will analyze the behavior of the sequence for both large and small values of n.

Behavior for Large n

As n becomes very large, an approaches 2/3. This suggests that for large n, the sequence will stay close to 2/3.

However, to provide a more rigorous approach, we can also consider the inequality:

[frac{-7}{2} leq frac{2n - 7}{3n^2} leq frac{2n}{3n^2} leq frac{2n}{3n^2} frac{2}{3}]

This inequality holds for all n ∈ ?, indicating that the sequence is indeed bounded.

Behavior for Small n

Let's also check the values of an for small values of n:

For n 1: [a_1 frac{2(1) - 7}{3(1)^2} frac{2 - 7}{3} frac{-5}{5} -1] For n 2: [a_2 frac{2(2) - 7}{3(2)^2} frac{4 - 7}{12} frac{-3}{12} -0.25] For n 3: [a_3 frac{2(3) - 7}{3(3)^2} frac{6 - 7}{27} frac{-1}{27} approx -0.037] For n 4: [a_4 frac{2(4) - 7}{3(4)^2} frac{8 - 7}{48} frac{1}{48} approx 0.021]

As we can see, the sequence takes values that are negative and close to zero, indicating that the sequence is bounded and converges to 2/3 as n approaches infinity.

Conclusion

Based on this analysis, we can conclude that the sequence:

-1 ≤ an ≤ (frac{2}{3})

Thus, the sequence (a_n frac{2n - 7}{3n^2}) is indeed bounded.