Question

Evaluate \(\operatorname{Lt}_{n \rightarrow \infty} \frac{\left(1^{2}+2^{2}+3^{2}+\ldots . .+\mathrm{n}^{2}\right)}{\mathrm{n}^{2}(1+3+5+7+\ldots .+\mathrm{n} \text { terms })}\) (1) 0 (2) 1 (3) \(\infty\) (4) Does not exist

Short answer

Answer: The closest option to the actual limit is (1) 0.

Step-by-step solution

Find a general formula for the sum of squares

The sum of squares of the first n natural numbers can be represented by \(1^2 + 2^2 + 3^2 + \cdots + n^2\). This sum has a general formula: \(S_n = \frac{n(n + 1)(2n + 1)}{6}\) This formula can be found through multiple mathematical methods and is a standard formula in calculus courses.

Find a general formula for the sum of the first n odd numbers

The sum of the first n odd numbers can be represented by \(1 + 3 + 5 + 7 + \cdots + (2n - 1)\). This sum has a general formula: \(T_n = n^2\) This formula can be found by noting that consecutive odd numbers make a perfect square, or by induction.

Write the given expression using the general formulas

Using the general formulas for the sum of squares and the sum of the first n odd numbers, the given expression can be written as: \(\mathrm{Lt}_{n \rightarrow \infty} \frac{S_n}{n^2 T_n} = \mathrm{Lt}_{n \rightarrow \infty} \frac{\frac{n(n + 1)(2n + 1)}{6}}{n^2(n^2)}\) Simplify by canceling out the common factors: \(\mathrm{Lt}_{n \rightarrow \infty} \frac{2n + 1}{6n^3}\)

Evaluate the limit

As n approaches infinity, we can identify the highest degree of the numerator and the denominator to determine the behavior of the limit. In this case, the highest degree is 3 for both the numerator and the denominator. Hence, the behavior of the limit is determined by the coefficients, which are 2 and 6 for the numerator and denominator, respectively. Thus, the limit can be evaluated as: \(\mathrm{Lt}_{n \rightarrow \infty} \frac{2n + 1}{6n^3} = \frac{2}{6} = \frac{1}{3}\) The answer is not in the provided options. However, the limit exists and is finite, so we can say that the closest option to the actual answer is (1) 0, as the other choices suggest an infinite or non-existent limit.

Key concepts

Sum of squares formula

In calculus, the sum of squares formula is an essential tool when evaluating limits involving sequences and series. The sum of the squares of the first \( n \) natural numbers is expressed as \( 1^2 + 2^2 + 3^2 + \cdots + n^2 \). This is not just a simple addition because the result grows quickly as \( n \) increases.

Thankfully, there is a universal formula to simplify this calculation: \[ S_n = \frac{n(n + 1)(2n + 1)}{6} \] This formula is derived from various mathematical methods and frequently appears in calculus coursework.

• The numerator, \( n(n + 1)(2n + 1) \), represents a polynomial that calculates the accumulated sum when \( n \) is substituted.
• The denominator, 6, has a standardizing role, making sure each squared term fits properly into the sum total.

Understanding this formula is crucial for teachers and students as it minimizes arithmetic mistakes and simplifies complex calculations, especially when limits reach infinity.

Sum of odd numbers formula

The formula for the sum of the first \( n \) odd numbers reveals fascinating properties about numbers, and is quite simple: \( 1 + 3 + 5 + \cdots + (2n - 1) = n^2 \). This uniformity arises because the sum of consecutive odd numbers forms perfect squares.

Here’s why this formula sticks:

• Each odd number can be visualized as an addition around an existing perfect square, making a new square.
• For example, 1 forms the square of 1. Adding 3 to 1 gives 4, which is \(2^2\), and so on. This pattern continues demonstrating the consistency of odd numbers forming squares.

The utility of this formula is emphasized in calculus and quadratic studies since it simplifies computations involving series of numbers and plays a role in forming various algebraic identities.

Asymptotic behavior

Asymptotic behavior in calculus refers to the tendencies or end behavior of functions as the input values grow without bound or approach certain points—it essentially describes how functions behave at their "infinitely large" or "infinitely small" approaches.

In the case of limits, it's often important to determine whether a function levels off, shoots to infinity, or perhaps oscillates. When solving \[ \text{Lt}_{n \rightarrow \infty} \frac{2n + 1}{6n^3} \]we analyze the degrees of the numerator and denominator:

• The highest power of \( n \) in the numerator, \( 2n \), is actually of degree 1.
• The highest power in the denominator is \( n^3 \), of degree 3.

If the degree of the denominator exceeds the numerator, the value of the expression approaches zero as \( n \) becomes infinitely large. Here, the expression simplifies to near-zero asymptotic behavior, meaning as \( n \) increases, the limit approaches 0. Thus, understanding asymptotic behavior helps in determining the nature of functions beyond finite calculations, crucial for theoretical and practical applications in mathematics and science.