Analyzing the Sequence ( C_n frac{2n^2 - 1}{4n^2} ): Boundedness and Monotonicity

Understanding the properties of a sequence, such as its boundedness and monotonicity, is crucial in many fields, including mathematics and computer science. In this article, we will explore the sequence [mathit{C}_n frac{2n^2 - 1}{4n^2}].

Step 1: Simplify the Sequence

Let's first simplify the given sequence expression to make the analysis easier:

[(mathit{C}_n) frac{2n^2 - 1}{4n^2}]

Simplifying the expression, we can separate the terms:

[(mathit{C}_n) frac{2n^2}{4n^2} - frac{1}{4n^2} frac{1}{2} - frac{1}{4n^2}]

Step 2: Check for Boundedness

A sequence is considered bounded if there exists a real number ( M ) such that for all ( n ), ( mathit{C}_n leq M ).

From the simplified expression, we have:

[(mathit{C}_n) frac{1}{2} - frac{1}{4n^2}]

As ( n ) approaches infinity, (frac{1}{4n^2}) approaches 0. Thus, the limit of the sequence as ( n ) goes to infinity is:

[lim_{n to infty} mathit{C}_n frac{1}{2} - 0 frac{1}{2}]

For all ( n geq 1 ), (mathit{C}_n ) is always greater than or equal to (frac{1}{2}) since (frac{1}{4n^2} ) is positive. Therefore, the sequence is bounded below by (frac{1}{2}).

Since the sequence approaches (frac{1}{2}) as ( n ) increases, it is also bounded above by (frac{1}{2}), making the sequence bounded.

Step 3: Check for Monotonicity

To determine if the sequence is monotonic, we need to check if (mathit{C}_{n 1} - mathit{C}_n) is positive, negative, or zero. If (mathit{C}_{n 1} - mathit{C}_n > 0), the sequence is strictly increasing. If (mathit{C}_{n 1} - mathit{C}_n

Let's calculate (mathit{C}_{n 1}) and then find (mathit{C}_{n 1} - mathit{C}_n):

[(mathit{C}_{n 1}) frac{2(n 1)^2 - 1}{4(n 1)^2}]

Expanding and simplifying:

[(mathit{C}_{n 1}) frac{2(n^2 2n 1) - 1}{4(n^2 2n 1)} frac{2n^2 4n 2 - 1}{4n^2 8n 4} frac{2n^2 4n 1}{4n^2 8n 4}]

Now, we compute (mathit{C}_{n 1} - mathit{C}_n):

[(mathit{C}_{n 1} - mathit{C}_n) frac{2n^2 4n 1}{4n^2 8n 4} - left(frac{1}{2} - frac{1}{4n^2}right)]

To simplify, we find a common denominator, which is ( 4n^2 8n 4 ):

[(mathit{C}_{n 1} - mathit{C}_n) frac{2n^2 4n 1}{4n^2 8n 4} - left(frac{2(4n^2 8n 4)}{2(4n^2 8n 4)} - frac{1}{4n^2 8n 4}right)]

Simplifying the expression further:

[(mathit{C}_{n 1} - mathit{C}_n) frac{2n^2 4n 1 - 8n^2 - 16n - 8 1}{4n^2 8n 4} frac{-6n^2 - 12n - 6}{4n^2 8n 4} frac{-6(n^2 2n 1)}{4(n^2 2n 1)} frac{-6}{4} -frac{3}{2}]

Since (mathit{C}_{n 1} - mathit{C}_n -frac{3}{2}0), this indicates that (mathit{C}_n) is strictly decreasing.

Conclusion

Boundedness:

The sequence (mathit{C}_n) is bounded specifically bounded below by (frac{1}{2}).

Monotonicity:

The sequence is strictly decreasing since (mathit{C}_{n 1} - mathit{C}_n

Thus, the sequence (mathit{C}_n) is bounded and strictly decreasing, indicating that it is effectively constant for very large ( n ).