Question

Consider the following sequences. a. Find the first four terms of the sequence. b. Based on part (a) and the figure, determine a plausible limit of the sequence. $$a_{n}=\frac{n^{2}}{n^{2}-1} ; n=2,3,4, \dots$$

Short answer

Answer: The first four terms of the sequence are $$a_2 =\frac{4}{3}, a_3 = \frac{9}{8}, a_4 = \frac{16}{15}, a_5 = \frac{25}{24}$$. The plausible limit of the sequence is 1.

Step-by-step solution

Find the first four terms of the sequence.

Use the formula for the sequence, $$a_n = \frac{n^2}{n^2 - 1}$$. For the first term, let n = 2: $$a_2 = \frac{2^2}{2^2 - 1} = \frac{4}{3}$$ For the second term, let n = 3: $$a_3 = \frac{3^2}{3^2 - 1} = \frac{9}{8}$$ For the third term, let n = 4: $$a_4 = \frac{4^2}{4^2 - 1} = \frac{16}{15}$$ For the fourth term, let n = 5: $$a_5 = \frac{5^2}{5^2 - 1} = \frac{25}{24}$$ So the first four terms of the sequence are $$a_2 =\frac{4}{3}, a_3 = \frac{9}{8}, a_4 = \frac{16}{15}, a_5 = \frac{25}{24}$$.

Determine a plausible limit of the sequence.

We notice that the terms of the sequence have the form ${ \frac{(n+1)(n-1) + 1}{(n+1)(n-1)}}$. Furthermore, we see that each denominator is just one less than the numerator, and as n grows larger, the difference between the numerator and denominator becomes less significant. Thus, it is plausible to conclude that the limit of the sequence is 1, since the ratio of the terms approaches 1 for larger n: $$\lim_{n\to\infty} \frac{n^2}{n^2 - 1} = 1$$ In conclusion, the first four terms of the sequence are: $$a_2 =\frac{4}{3}, a_3 = \frac{9}{8}, a_4 = \frac{16}{15}, a_5 = \frac{25}{24}$$, and the plausible limit of the sequence is 1.

Key concepts

sequences and series

A sequence is a list of numbers arranged in a specific order. Each number in a sequence is called a term. In mathematics, sequences are a fundamental concept as they can describe an ordered collection of objects, such as the natural numbers or even more complicated formulations. Sequences can be finite, with a certain number of terms, or infinite, continuing indefinitely. In the original exercise, we look at the sequence given by the formula \( a_n = \frac{n^2}{n^2 - 1} \). To find the terms of a sequence, substitute integer values of \( n \) into the sequence’s formula.

• The first four terms found using \( n = 2, 3, 4, \) and \( 5 \) are \( \frac{4}{3}, \frac{9}{8}, \frac{16}{15}, \frac{25}{24} \).

A series, on the other hand, is what you get when you add up the terms of a sequence. While the exercise focuses on the sequence itself, understanding series is essential for topics like calculus, where you might need to add an infinite number of terms. Each series can be described as either convergent, meaning it approaches a specific value as more terms are added, or divergent, where it does not have a finite limiting value.

convergence

Convergence is a key idea when studying sequences and series. A sequence converges if its terms approach a specific value, known as the limit, as \( n \) becomes very large. In simple terms, if the terms of the sequence settle down to a particular number as you move further and further down the sequence, it's said to "converge."
In the original exercise, the terms of the sequence \( a_n = \frac{n^2}{n^2 - 1} \) approach the value 1 as \( n \) increases.

• For small values of \( n \), the numerator and denominator differ slightly, but as \( n \) grows, the impact of subtracting 1 from \( n^2 \) becomes negligible.
• Thus, the terms get closer and closer to 1, indicating that the sequence converges to 1.

Understanding convergence is crucial not only in sequences but also in series. With series, just like sequences, convergence means that as you add more terms, the total sum approaches a specific number. This concept is foundational in many areas of mathematics and essential for understanding the behavior of functions and their approximations.

limit of a sequence

The limit of a sequence is the value that the terms of the sequence get closer to as the sequence progresses towards infinity. Calculating the limit involves determining the long-term behavior of the sequence, and it helps in predicting what the terms approach.
In our example, the sequence \( a_n = \frac{n^2}{n^2 - 1} \) has a limit of 1.

• As \( n \) grows very large, the effect of that -1 in the denominator vanishes relative to \( n^2 \), leading the fraction \( \frac{n^2}{n^2 - 1} \) closer to \( \frac{n^2}{n^2} = 1 \).

To find the limit, you often look for patterns or behaviors as \( n \) becomes large. Advanced methods such as L'Hôpital's Rule or manipulating expressions algebraically can also be used to determine limits.
Knowing the limit of a sequence is valuable in fields like calculus and analysis, where understanding the limiting behavior of functions and sums can provide deep insights into continuity, differentiability, and integrals.