### Problem Statement**Find the limit of \( s(n) \) as \( n \to \infty \).**### Function Definition\[ s(n) = \frac{1}{n^2} \left[ \frac{n(n + 1)}{9} \right] \]### ExplanationThe equation provided defines the function \( s(n) \) in terms of \( n \). The function includes a term \( n(n + 1) \) which is first divided by 9, and the result is then multiplied by the reciprocal of \( n^2 \).### Calculating the LimitThe goal is to determine the behavior of \( s(n) \) as \( n \) approaches infinity. Consider applying limit laws and algebraic manipulation to evaluate the limit: 1. **Substitute and Simplify:** \[ \text{Inside the brackets: } \frac{n(n + 1)}{9} \approx \frac{n^2}{9} + \frac{n}{9} \]2. **Calculate the Limit:** \[ s(n) = \frac{1}{n^2} \left( \frac{n^2}{9} + \frac{n}{9} \right) \] \[ s(n) = \frac{1}{9} + \frac{1}{9n} \]3. **Final Limit:** \[ \lim_{n \to \infty} s(n) = \lim_{n \to \infty} \left( \frac{1}{9} + \frac{1}{9n} \right) = \frac{1}{9} + 0 = \frac{1}{9} \]Thus, the limit of \( s(n) \) as \( n \) approaches infinity is \( \frac{1}{9} \).

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Answered: Find the limit of s(n) as n → ∞0. 1…

Math

Calculus

Find the limit of s(n) as n → ∞0. 1 [n(n+1 1) n² 2 9 s(n) :

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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