Problem 4 - Calculus for Biology and Medicine

Question

Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ f(n)=\frac{1}{1+n^{2}} $$

Step-by-Step Solution

Answer

The sequence values are: \( a_0 = 1, a_1 = \frac{1}{2}, a_2 = \frac{1}{5}, a_3 = \frac{1}{10}, a_4 = \frac{1}{17}, a_5 = \frac{1}{26} \).

Step 1: Identify the Sequence Formula

The sequence is defined by the function \( f(n) = \frac{1}{1+n^2} \). We need to calculate \( a_n = f(n) \) for each value of \( n \) from 0 to 5.

Step 2: Calculate for n=0

Substitute \( n = 0 \) into the formula: \[ a_0 = f(0) = \frac{1}{1+0^2} = \frac{1}{1} = 1 \] Thus, \( a_0 = 1 \).

Step 3: Calculate for n=1

Substitute \( n = 1 \) into the formula: \[ a_1 = f(1) = \frac{1}{1+1^2} = \frac{1}{2} \] Thus, \( a_1 = \frac{1}{2} \).

Step 4: Calculate for n=2

Substitute \( n = 2 \) into the formula: \[ a_2 = f(2) = \frac{1}{1+2^2} = \frac{1}{5} \] Thus, \( a_2 = \frac{1}{5} \).

Step 5: Calculate for n=3

Substitute \( n = 3 \) into the formula: \[ a_3 = f(3) = \frac{1}{1+3^2} = \frac{1}{10} \] Thus, \( a_3 = \frac{1}{10} \).

Step 6: Calculate for n=4

Substitute \( n = 4 \) into the formula: \[ a_4 = f(4) = \frac{1}{1+4^2} = \frac{1}{17} \] Thus, \( a_4 = \frac{1}{17} \).

Step 7: Calculate for n=5

Substitute \( n = 5 \) into the formula: \[ a_5 = f(5) = \frac{1}{1+5^2} = \frac{1}{26} \] Thus, \( a_5 = \frac{1}{26} \).

Key Concepts

Recursive Sequences

Recursive sequences are a fascinating part of mathematics. They allow us to define sequences where each term is determined based on the previous one or several terms. However, unlike recursive sequences, the calculations in our exercise were not based on the preceding value but on a direct mathematical function. Recursive sequences often look like this:

• The first term, denoted as \( a_0 \), is provided or assumed.
• Each subsequent term \( a_n \) is defined based on one or more of its preceding terms, such as \( a_{n} = a_{n-1} + a_{n-2} \).
• Common examples are Fibonacci and arithmetic sequences.

In the original exercise, our sequence used a different approach by employing the given function \( f(n) = \frac{1}{1+n^2} \). This function allowed each term \( a_n = f(n) \) to be calculated independently, based on the input \( n \). While the sequence was not recursive in this case, understanding both recursive and non-recursive forms can deepen your understanding of how sequences operate.

Mathematical Functions

Mathematical functions are like machines where you input a number, and out comes another number based on specific rules. They play a crucial role in defining sequences or series. In the exercise, the function \( f(n) = \frac{1}{1+n^2} \) determines each sequence value.Here’s how the process works:

• Input an integer \( n \) representing the position in the sequence.
• Apply the function by substituting \( n \) into \( \frac{1}{1+n^2} \).
• Compute the result to get the sequence term \( a_n \).

For example, with \( n=0 \), the term is calculated as \( a_0 = \frac{1}{1+0^2} = 1 \). Functions help not only in finding terms in a sequence but also in analyzing properties like growth, limits, and patterns. By practicing with sequences and functions, you refine your ability to decipher complex mathematical relationships.

Calculus Concepts

Calculus might seem a stretch from solving simple sequences, but understanding sequences leads directly into many core concepts of calculus. Sequences, in fact, are the building blocks for series, and both are integral parts of calculus.Let's explore their role:

• Limits: In calculus, we often explore the idea of what happens as \( n \) approaches infinity (a focus for future calculus studies).
• Convergence: This concerns whether a sequence or series settles on a specific value as numbers get larger. In our example, as \( n \) increases, \( a_n \) approaches zero.
• Understanding behavior: Sequences allow us to predict behavior in more complex calculus functions.

In our sequence, each term \( a_n = \frac{1}{1+n^2} \) gradually gets smaller as \( n \) gets larger. This trend helps illustrate how sequences bring insights into calculus concepts like limits and convergence, paving the way for deeper studies in the field. Grasping these basics provides a solid grounding for tackling more advanced calculus problems.