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Chapter 9: Problem 1

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{n}{3 n-1}\)

Short Answer

Expert verified
The sequence converges to \( \frac{1}{3} \).

Step by step solution

01

Write down the explicit formula

The given explicit formula for the sequence is \( a_{n} = \frac{n}{3n - 1} \). This means each term of the sequence is calculated by substituting \( n \) with its respective sequence position number.
02

Calculate the first five terms

To find the first five terms of the sequence, substitute \( n = 1, 2, 3, 4, \) and \( 5 \) into the formula \( a_{n} = \frac{n}{3n - 1} \):- \( a_{1} = \frac{1}{3\cdot1 - 1} = \frac{1}{2} \)- \( a_{2} = \frac{2}{3\cdot2 - 1} = \frac{2}{5} \)- \( a_{3} = \frac{3}{3\cdot3 - 1} = \frac{3}{8} \)- \( a_{4} = \frac{4}{3\cdot4 - 1} = \frac{4}{11} \)- \( a_{5} = \frac{5}{3\cdot5 - 1} = \frac{5}{14} \) Hence, the first five terms are \( \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, \frac{5}{14} \).
03

Determine if the sequence converges or diverges

To determine if the sequence converges or diverges, evaluate \( \lim_{n \to \infty} a_{n} \). Simplify the expression \( \lim_{n \to \infty} \frac{n}{3n - 1} \):Divide both the numerator and the denominator by \( n \):\[ \lim_{n \to \infty} \frac{\frac{n}{n}}{\frac{3n - 1}{n}} = \lim_{n \to \infty} \frac{1}{3 - \frac{1}{n}} \]As \( n \to \infty \), \( \frac{1}{n} \to 0 \). Thus, the limit becomes:\[ \lim_{n \to \infty} \frac{1}{3 - 0} = \frac{1}{3} \]Therefore, the sequence converges to \( \frac{1}{3} \).
04

Conclusion

The sequence \( a_{n} = \frac{n}{3n - 1} \) has its first five terms as \( \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, \frac{5}{14} \). It converges to \( \frac{1}{3} \) as \( n \to \infty \).

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Key Concepts

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

Explicit Formula for Sequences
The explicit formula is like a magic key for sequences. It allows us to find any term in the sequence by simply plugging the term's position number into the formula. For example, given \( a_{n} = \frac{n}{3n - 1} \), this formula lets us calculate each term by substituting \( n \) with the specific term number. This means:
  • For \( n = 1 \), we find the first term \( a_{1} = \frac{1}{2} \).
  • For \( n = 2 \), the second term \( a_{2} = \frac{2}{5} \), and so on.
Using an explicit formula is a straightforward way to quickly uncover as many terms as needed without having to build the sequence one term at a time from the beginning.
Limit of a Sequence
The limit of a sequence helps us understand the sequence's behavior as it continues indefinitely. Specifically, the limit investigates what value the terms of the sequence approach as \( n \, \) (the term number) becomes very large or tends to infinity.
In the sequence \( a_{n} = \frac{n}{3n - 1} \), as \( n \) increases, both the numerator and denominator grow, but the trick is observing the ratio.Using a smart algebraic move, we divide every part of the fraction by \( n \). This results in simplifying our expression to \( \frac{1}{3 - \frac{1}{n}} \).
Due to \( \frac{1}{n} \) approaching 0 as \( n \) approaches infinity, the whole fraction converges to \( \frac{1}{3} \).
Thus, we conclude that the sequence converges to this limit.
Divergence and Convergence
Divergence and convergence describe the ending behavior of a sequence. They provide insight into whether a sequence settles down to a specific value or not.
  • A sequence converges if it approaches a single finite limit, as is the case with \( a_{n} = \frac{n}{3n - 1} \), where it reaches \( \frac{1}{3} \).
  • Conversely, divergence indicates that the terms of a sequence do not settle into any particular value, and instead, they may grow without bound or oscillate.
Determining whether a sequence converges or diverges can reveal much about its long-term behavior and stability.
Substitution in Sequences
Substitution is a vital step when working with sequences, especially when we're using an explicit formula. It involves replacing the variable \( n \) with specific numbers to calculate the sequence terms.
For instance, using \( a_{n} = \frac{n}{3n - 1} \), by substituting \( n = 1 \), we get \( a_{1} = \frac{1}{2} \). Similarly, for \( n = 2 \), we find \( a_{2} = \frac{2}{5} \). This process allows us to determine exactly how the initial terms of the sequence look like.
This practice is not only useful for writing out the first few terms but also essential for exploring and analyzing the general behavior of the sequence.

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Most popular questions from this chapter

In Problems 21-30, find an explicit formula \(a_{n}=\) for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). \(1-\frac{1}{2}, \frac{1}{2}-\frac{1}{3}, \frac{1}{3}-\frac{1}{4}, \frac{1}{4}-\frac{1}{5}, \ldots\)

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{(x-2)^{n}}{n} $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=(2 n)^{1 / 2 n}\)

One can sometimes find a Maclaurin series by the method of equating coefficients. For example, let $$ \tan x=\frac{\sin x}{\cos x}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots $$ Then multiply by \(\cos x\) and replace \(\sin x\) and \(\cos x\) by their series to obtain $$ \begin{aligned} x-\frac{x^{3}}{6}+\cdots &=\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\left(1-\frac{x^{2}}{2}+\cdots\right) \\ &=a_{0}+a_{1} x+\left(a_{2}-\frac{a_{0}}{2}\right) x^{2}+\left(a_{3}-\frac{a_{1}}{2}\right) x^{3}+\cdots \end{aligned} $$ Thus, $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}-\frac{a_{0}}{2}=0, \quad a_{3}-\frac{a_{1}}{2}=-\frac{1}{6}, \quad \ldots $$ so $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}=0, \quad a_{3}=\frac{1}{3}, \quad \ldots $$ and therefore $$ \tan x=0+x+0+\frac{1}{3} x^{3}+\cdots $$ which agrees with Problem 1. Use this method to find the terms through \(x^{4}\) in the series for \(\sec x\).

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems 43-48 to see that you get the same answers using the methods of Section 9.7. $$ (\sin x) /(\exp x) $$
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