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Chapter 9: Problem 37

Find the limit (if possible) of the sequence. $$ a_{n}=\frac{5 n^{2}}{n^{2}+2} $$

Short Answer

Expert verified
The limit of the sequence \( a_{n}=\frac{5 n^{2}}{n^{2}+2} \) as \( n \) tends to infinity is 5.

Step by step solution

01

Identify the form of the sequence

The sequence \( a_{n}=\frac{5 n^{2}}{n^{2}+2} \) is given by a rational function. When the degrees of the numerator and the denominator are the same, the limit of the ratio as \( n \) goes to infinity is the ratio of the leading coefficients.
02

Compute the limit

As n tends to infinity, the limit of the sequence can be approximated to be the ratio of the coefficients of the highest degree of \( n \) as fractions in the numerator and denominator. Thus, the limit \( L \) is \( L = \frac{5}{1} = 5 \).
03

Provide the final answer

After evaluating the limit, we conclude that as \( n \) tends to infinity, the sequence \( a_{n}=\frac{5 n^{2}}{n^{2}+2} \) tends to 5. This is the limiting value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Every rational function is a fraction where both the numerator and the denominator are polynomials. These are powerful tools in mathematics because they describe how quantities vary relative to each other. In the given sequence, \( a_n = \frac{5 n^2}{n^2 + 2} \), the function is rational because it is the ratio of two polynomials. The numerator is \( 5n^2 \), and the denominator is \( n^2 + 2 \). You'll notice both the numerator and denominator have the same polynomial degree, which is 2. This property simplifies the process of determining limits as it becomes just about comparing leading coefficients.
Leading Coefficients
Leading coefficients in a polynomial are the coefficients of the term with the highest degree. When analyzing limits of rational functions where the degrees of the numerator and the denominator are the same, it is crucial to examine their leading coefficients. For this function, \( 5n^2 \) in the numerator has a leading coefficient of 5, and \( n^2 + 2 \) in the denominator has a leading coefficient of 1. To find the limit as \( n \) approaches infinity, we take these leading coefficients and form a ratio: \( \frac{5}{1} = 5 \). This ratio directly tells us the limit of the sequence, showcasing a simple yet effective method of solving many rational function limits.
Infinite Limits
Infinite limits help describe the behavior of functions as they approach very large values. This concept is vital in understanding sequences like the one in our problem. As \( n \) becomes very large, the lower order terms (like the constant 2 in the denominator) become negligible compared to the terms with the highest degree (like \( n^2 \)). Thus, the sequence simplifies primarily to a comparison of the highest degree terms: \( 5n^2 \) and \( n^2 \). Therefore, the infinite limit of this sequence as \( n \) grows indefinitely is determined by the ratio of these leading terms, which results in a limiting value of 5. Understanding infinite limits allows one to predict long-term behavior of sequences and functions, which is both a crucial and often-used skill in calculus.

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Most popular questions from this chapter

Let \(f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{(2 n+1) !}\) and \(g(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !} .\) (a) Find the intervals of convergence of \(f\) and \(g\). (b) Show that \(f^{\prime}(x)=g(x)\). (c) Show that \(g^{\prime}(x)=-f(x)\). (d) Identify the functions \(f\) and \(g\).

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2 n+1}, \quad-1 \leq x \leq 1 $$

For \(n>0\), let \(R>0\) and \(c_{n}>0 .\) Prove that if the interval of convergence of the series \(\sum_{n=0}^{\infty} c_{n}\left(x-x_{0}\right)^{n}\) is \(\left(x_{0}-R, x_{0}+R\right]\), then the series converges conditionally at \(x_{0}+R\).

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Consider the sequence \(\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}\), for \(n \geq 2\). (c) Find lim \(a_{n}\).

Find all values of \(x\) for which the series converges. For these values of \(x\), write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$
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