Question

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. $$a_{n}=\frac{(n-1)^{2}}{\left(n^{2}-1\right)} ; n=2,3,4, \dots$$

Short answer

Answer: 1

Step-by-step solution

Calculate the terms of the sequence.

Using the formula \(a_n = \frac{(n-1)^2}{(n^2-1)}\), calculate the first 10 terms of the sequence. n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 --|---|----|----|----|----|-----|-----|-----|-----|----- an| 1 |3/4 |2/3 |3/5 |2/5 |5/12 |6/19 |5/14 |4/21 |5/22

Observe the behavior of the terms as n increases.

As we look at the terms in the table, we can see that they seem to decrease as n gets larger. It suggests that the terms are getting closer to some limiting value, possibly zero.

Determine a plausible value for the limit of the sequence or state that it does not exist.

As n goes to infinity, the sequence becomes $$ \lim_{n\to\infty} \frac{(n-1)^2}{(n^2-1)} = \lim_{n\to\infty} \frac{n^2 - 2n + 1}{n^2 - 1}$$ For large n, the quadratic terms (\(n^2\)) dominate the expression, and we can ignore the lower-degree terms. The expression then simplifies to $$ \lim_{n\to\infty} \frac{n^2}{n^2} = 1$$ Thus, a plausible value for the limit of the sequence is 1, as the terms seem to approach 1 as n gets larger.

Key concepts

Calculus

Calculus is a branch of mathematics that explores how things change. At its heart are derivatives and integrals. These are tools that help understand how quantities grow, shrink, or move with respect to one another. In the context of sequences, calculus might not always come into play directly, but the concept of limits, which is a fundamental idea in calculus, is vital. The steps in the limit process involve evaluating how a sequence behaves as the index increases, often towards infinity, thereby studying the 'end behavior' of mathematical objects.

Sequences and Series

In mathematics, sequences are ordered lists of numbers. Each number in the sequence is called a term. The sequences can be finite or infinite, depending on whether they terminate after a certain number of terms or continue indefinitely. Understanding sequences involves looking at the pattern or rule that defines how the terms are generated.
A series is the sum of the terms of a sequence. In this exercise, though we're dealing with sequences, the concepts overlap, especially when considering converging and diverging behaviors. Observing the list of terms calculated, one sees how it changes with each step, leading to conclusions about patterns or trends. This is part of analyzing sequences.

Limit of a Sequence

A limit of a sequence is what the terms of the sequence approach as the sequence progresses indefinitely. More formally, a sequence \( a_n \) has a limit \( L \) if for every positive number \( \epsilon \), there is a point beyond which all the sequence's terms \( a_n \) lie within the distance \( \epsilon \) from \( L \). This concept helps in understanding the long-term behavior of sequences.
When analyzing the sequence given by \( a_n = \frac{(n-1)^2}{n^2 - 1} \), the step-by-step solution demonstrates looking at how the sequence behaves as \( n \) increases. Large terms in the denominator and numerator are compared to deduce that the limit is 1, as the lower-degree terms become insignificant. Thus, through simplifying and reasoning, the limit gives a valuable insight into the sequence's behavior at infinity.