Problem 39 - Precalculus with Limits

Question

In Exercises 39-48, write the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, explain why. Assume \(n\) begins with 1. $$ a_n = \dfrac{n+1}{n^2 +1} $$

Step-by-Step Solution

Answer

The first five terms of the sequence are 1, \(3/5\), \(4/10\), \(5/17\), \(6/26\). The limit of the sequence \( a_n = \dfrac{n+1}{n^2 +1} \) is 0 because the denominator \(n^2+1\) will grow much faster than the numerator \(n+1\) as \(n\) approaches infinity.

Step 1: Write the first five terms of the sequence

To find these, substitute the values \(n = 1, 2, 3, 4, 5\) into the equation \( a_n = \dfrac{n+1}{n^2 +1} \).\n \(a_1 = \dfrac{1+1}{1^2+1} = \dfrac{2}{2} = 1 \)\n \(a_2 = \dfrac{2+1}{2^2+1} = \dfrac{3}{5} \)\n \(a_3 = \dfrac{3+1}{3^2+1} = \dfrac{4}{10} = 0.4 \)\n \(a_4 = \dfrac{4+1}{4^2+1} = \dfrac{5}{17} \approx 0.294\)\n \(a_5 = \dfrac{5+1}{5^2+1} = \dfrac{6}{26} = 0.231 \)

Step 2: Check if highest powers in sequence formula are the same

In both the numerator and denominator, there is a highest power of \(n\). In the numerator, it's \(n^1\) and in the denominator, it's \(n^2\). Since the highest power in the denominator is greater, this means the limit as \(n\) approaches infinity will be 0.

Step 3: Find the limit of the sequence

Now, we will find the limit of the sequence \(a_n\). To find the limit of the sequence, we can simplify the formula and see what it approaches as \(n\) approaches infinity. In this case, the denominator \(n^2+1\) will grow much faster than the numerator \(n+1\). As a result, as \(n\) approaches infinity, the entire value of \(a_n\) will approach 0. This means the limit of this sequence is 0.

Key Concepts

Sequences in Precalculus

In precalculus, sequences are an essential concept, often introduced as functions defined on the set of natural numbers. They serve as a foundational block for understanding how various number patterns behave as their indices increase.
Each term in a sequence is typically expressed in terms of its position, known as its index, denoted by \( n \), and a formula is provided to calculate the term's value. For instance, in the given exercise, the sequence \( a_n \) is defined by the formula \( a_n = \frac{n+1}{n^2 +1} \).
To get a grasp of this sequence, we begin by generating its first few terms. This involves plugging in the initial natural numbers into the formula, starting with \( n = 1 \) and continuing sequentially. Understanding these early terms helps to recognize patterns and inform us about the sequence's behavior as \( n \) grows larger, which is a crucial step toward discerning its limit.

Convergence of Sequences

The convergence of sequences is about determining if a sequence tends towards a specific value as its index \( n \) increases indefinitely. A sequence that has such a definite value is said to be convergent, and the value it approaches is called its limit.
When analyzing convergence, we look for the dominant term in the sequence's formula, especially when \( n \) gets very large. For the exercise's sequence \( a_n \) the terms in the denominator \( n^2 \) become significantly larger than those in the numerator \( n \) as \( n \) approaches infinity. This suggests that the sequence's terms will get closer and closer to zero. Therefore, we can make a provisional conclusion that the sequence is convergent and that its limit is likely to be 0.
However, it is important to formally prove such a claim using the definition of limit and sometimes employing additional mathematical tools like the Squeeze theorem or L'Hôpital's rule, which are beyond the scope of precalculus but are central in calculus.

Limits in Mathematics

Limits are a fundamental concept in mathematics, particularly in calculus. They describe the value that a function or a sequence approaches as the input or index gets arbitrarily close to some point.
In the context of sequences, as shown in the exercise, we investigate what happens to \( a_n \) as \( n \) becomes very large. If there exists a number \( L \) such that the sequence’s terms can get arbitrarily close to \( L \) for all sufficiently large \( n \) then we say that the limit of \( a_n \) as \( n \) approaches infinity is \( L \). Mathematically, this is denoted as \( \lim_{{n\to\infty}} a_n = L \).
For the given sequence, by comparing the degree of \( n \) in the numerator and denominator, we conclude that the limit is 0 because the terms are getting smaller as \( n \) gets larger. This is a common outcome for sequences where the degree of \( n \) in the denominator is higher than that in the numerator — a crucial insight when determining limits algebraically.