Problem 13 - Calculus for Biology and Medicine

Question

Write each sum in expanded form. $$ \sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2} \frac{1}{n} $$

Step-by-Step Solution

Answer

The sum in expanded form is \( \frac{1}{n^3} + \frac{4}{n^3} + \cdots + \frac{n^2}{n^3} \).

Step 1: Understanding the Sigma Notation

The expression \( \sum_{k=1}^{n} \left( \frac{k}{n} \right)^{2} \frac{1}{n} \) uses sigma notation to represent the sum of a series of terms for \( k \) ranging from 1 through \( n \). This means we are systematically substituting values of \( k \) from 1 to \( n \) into the expression \( \left( \frac{k}{n} \right)^{2} \frac{1}{n} \) and adding the results together.

Step 2: Substituting Values

For each integer \( k \) going from 1 to \( n \), substitute that value into the expression \( \left( \frac{k}{n} \right)^{2} \frac{1}{n} \). This means when \( k = 1 \), the term is \( \left( \frac{1}{n} \right)^{2} \frac{1}{n} \). Similarly, for \( k = 2 \), the term is \( \left( \frac{2}{n} \right)^{2} \frac{1}{n} \), continuing up to \( k = n \).

Step 3: Writing in Expanded Form

Write out each term of the sequence explicitly, as derived in the previous step. Thus, the expanded form of the sum is: \[ \left( \frac{1}{n} \right)^{2} \frac{1}{n} + \left( \frac{2}{n} \right)^{2} \frac{1}{n} + \cdots + \left( \frac{n}{n} \right)^{2} \frac{1}{n}. \] This results in \[ \frac{1}{n^3} + \frac{4}{n^3} + \cdots + \frac{n^2}{n^3}. \]

Key Concepts

Expanded Form

When working with sigma notation, it is often useful to convert that notation into expanded form. An expanded form is the complete, detailed expression of each term in the series, laid out in sequence. In our exercise, we start with the sigma notation:

• \(\sum_{k=1}^{n} \left(\frac{k}{n}\right)^{2} \frac{1}{n}\)

This notation represents the sum of terms from \(k = 1\) to \(k = n\). To write this in expanded form, we replace the sigma notation with an enumeration of each term:

• For \(k = 1\): \(\left(\frac{1}{n}\right)^{2} \frac{1}{n}\)
• For \(k = 2\): \(\left(\frac{2}{n}\right)^{2} \frac{1}{n}\)
• ...
• For \(k = n\): \(\left(\frac{n}{n}\right)^{2} \frac{1}{n}\)

This results in the sequence we know as expanded form:

• \(\left(\frac{1}{n}\right)^{2} \frac{1}{n} + \left(\frac{2}{n}\right)^{2} \frac{1}{n} + \cdots + \left(\frac{n}{n}\right)^{2} \frac{1}{n}\)

To simplify further, this is written as:

• \(\frac{1}{n^3} + \frac{4}{n^3} + \cdots + \frac{n^2}{n^3}\)

Each fraction represents one part of the whole sum, showing clearly how they build to the result.

Series Summation

Series summation is the process of adding a sequence of numbers, where each number is defined by a specific rule or formula. In our exercise, we're summing a sequence given by:

• \(\sum_{k=1}^{n} \left(\frac{k}{n}\right)^{2} \frac{1}{n}\)

This entails plugging values for \(k\) ranging from 1 to \(n\) into the specified formula and then summing the resulting terms. The idea is to systematically replace \(k\) into the expression and observe the pattern:

• The terms look like this: \(\frac{k^2}{n^3}\)

In the end, we are adding up all values from \(k = 1\) to \(k = n\) to give a single sum:

• \(\frac{1}{n^3} + \frac{4}{n^3} + \cdots + \frac{n^2}{n^3}\)

Series summation is a fundamental concept in calculus because it helps us understand calculative processes that are essential for differentiating and integrating functions, particularly in biological systems.

Calculus for Biology

Calculus often plays a crucial role in biology, helping to model the natural world and solve complex problems. Specifically, the series summation illustrated in our exercise has practical applications in biological research. Such sums can be used to approximate areas, volumes, or changes within biological systems.

Biological Models and Calculus

In biology, calculus helps model population growth, the spread of diseases, or changes in an environment over time. By understanding series summation, biologists can better predict and track these processes, using the sums of series to approximate biological phenomena.

Applications of the Continued Summation

In calculus, summation is used to approximate integrals, which describe total quantities (such as total population) or changes (such as temperature variation in a habitat) over a given time.

• Approximating changes in population size over time using discrete time intervals.
• Estimating resource consumption or disease spread in an ecosystem.

Through these methods, calculus provides crucial insights into dynamic and complex biological systems, transforming broad concepts into precise calculations and forecasts.