give the calculation step by step as like as full answer in 2 steps and explanation block in every steps.
And final answer summary
Definition of Parameters
Solutions are reviewed by our quality t

Question

give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. 
And final answer summary
Definition of Parameters 
Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. 
Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. 
Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. 
Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. 
Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. 
Structure: Structure refers to how the solution is organized and presented in relation to the requi

VIPSolutions · Accepted Answer

**Definition of Parameters**  
- **Relevancy:** Alignment of the solution with the question asked.  
- **Completeness:** Thoroughness in addressing all aspects including calculations and explanations.  
- **Accuracy:** Correctness of the mathematical process and final result.  
- **Clarity:** Ease of understanding, use of simple language, and avoidance of unnecessary complexity.  
- **Structure:** Logical organization and stepwise presentation.  
- **Voice:** Human-like, clear, and concise tone.

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**Step 1: Express the Sequence and Analyze Its Form**

Given sequence:  
$$ a_n = \frac{5n^2 + 7}{(n+1)^3} $$

**Explanation:**  
The given sequence is a rational function where the numerator is a quadratic polynomial and the denominator is a cubic polynomial. The degree of the denominator (3) is higher than that of the numerator (2).

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**Step 2: Find the Limit as $ n \to \infty $**

Rewrite the sequence by dividing numerator and denominator by $ n^3 $:  
$$
a_n = \frac{5n^2 + 7}{(n+1)^3} = \frac{5n^2/n^3 + 7/n^3}{(n+1)^3/n^3}
= \frac{5/n + 7/n^3}{(1 + 1/n)^3}
$$

As $ n \to \infty $:  
- $ 5/n \to 0 $
- $ 7/n^3 \to 0 $
- $ (1 + 1/n)^3 \to 1 $

So,  
$$
\lim_{n \to \infty} a_n = \frac{0 + 0}{1} = 0
$$

**Explanation:**  
Both terms in the numerator approach zero much faster than the denominator grows. Thus, the whole sequence approaches zero as $ n $ becomes very large.

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**Final Answer Summary**

**The given sequence converges. Its limit is $ 0 $ as $ n \to \infty $, since the degree of the denominator exceeds the degree of the numerator, causing the terms to shrink towards zero.**

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