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A sequence is a list of numbers arranged in a specific order. In our example, the sequence is described by the formula \( a_{n} = \frac{n^{2} + 5}{n^{2} + 1} \). Sequences can either converge to a limit or diverge. When sequences have a limit, they approach a specific value as \( n \) grows larger.
A series is the sum of the terms of a sequence. However, in the context of this exercise, we're more interested in observing how individual terms in a sequence unfold. For the given sequence, each term is generated by substituting \( n \) with consecutive natural numbers, i.e., \( n = 0, 1, 2, 3, \ldots \).
When we calculate the first terms, we encounter:

• \(a_0 = 5\)
• \(a_1 = 3\)
• \(a_2 = 1.8\)
• \(a_3 = 1.4\)
• \(a_4 \approx 1.235\)

We are noticing how each term substantially decreases as \( n \) increases. This frequent feature of sequences in calculus helps in predicting behavior at large \( n \) values.

Calculus is an invaluable tool in biology, providing methods to model and analyze dynamic systems. Sequences and limits are crucial in understanding processes like population growth, enzyme reactions, and more.
In biology, an understanding of limits helps explain how certain biological systems stabilize over time. This is similar to finding the limit of a sequence in calculus.
For instance, consider a population that grows according to a particular pattern. Initially, growth occurs rapidly but eventually levels off as resources become limited.

• This leveling off behavior mirrors what we found in the exercise: as \( n \rightarrow \infty \), the sequence approaches its limit of 1.
• Biologically, this model might predict populations reaching a stable size when growth factors match constraints.

Using calculus, biologists can gain insights into such models and make informed predictions.

Infinite limits deal with the behavior of functions as they extend towards infinity, either in their domain or range. In sequences, as \( n \) approaches infinity, we investigate whether terms tend to stabilize at a certain value.
Calculating the limit of the sequence \( a_{n} = \frac{n^{2} + 5}{n^{2} + 1} \) involved comparing dominant terms:

• The terms \(\frac{5}{n^{2}}\) and \(\frac{1}{n^{2}}\) vanish as \(n\) becomes very large.
• The higher degree terms \( n^2 \) in numerator and denominator dominate, simplifying our task to looking at \(\frac{1 + \frac{5}{n^{2}}}{1 + \frac{1}{n^{2}}} = 1\) as \(n\) approaches infinity.

Understanding these concepts is key to solving complex calculus problems, especially those involving real-world applications where determining behavior notably "in the long run" is required.