Problem 16 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. $$ \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}} $$

Step-by-Step Solution

Answer

\( \sum_{n=1}^{4} \frac{1}{\sqrt{n}} \)

Step 1: Identify the pattern

We first look at the terms of the sequence: \( \frac{1}{\sqrt{1}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{4}} \). Each term follows a specific pattern that can be written in the form \( \frac{1}{\sqrt{n}} \) where \( n \) represents the term index.

Step 2: Determine the range of the summation index

Next, we observe how the sequence indexes map to the terms. The first term corresponds to \( n=1 \), the second term to \( n=2 \), and so on. This indicates that \( n \) starts at 1 and ends at 4.

Step 3: Write the sequence in sigma notation

Using the pattern \( \frac{1}{\sqrt{n}} \) and the index range from 1 to 4, we can write the original series in sigma notation as: \[ \sum_{n=1}^{4} \frac{1}{\sqrt{n}} \]

Key Concepts

Sigma Notation

In calculus and mathematics, **Sigma Notation** is a way of expressing the summation of terms in a sequence using the Greek letter Sigma (Σ). Sigma notation provides a compact way to represent long sums and is often used in series and sequence problems. The notation takes the form \[ \sum_{i=a}^{b} f(i) \] where:

• \( i \) is the summation index (the variable that changes).
• \( a \) is the lower bound of the summation.
• \( b \) is the upper bound of the summation.
• \( f(i) \) is the function of \( i \) that will be summed over.

For example, if you have the series \( \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}} \), it translates to:\[ \sum_{n=1}^{4} \frac{1}{\sqrt{n}} \]Here, each term of the sequence is represented by the function \( \frac{1}{\sqrt{n}} \), where \( n \) takes on each integer value from 1 to 4. It's a concise and efficient way to express sums, especially in calculus.

Sequences

**Sequences** are ordered lists of numbers that often follow a specific rule or pattern. Each number in the sequence is termed an element or term. Sequences are crucial in calculus and the study of series, acting as the foundation for further calculations and analysis. There are different types of sequences, including:

• Arithmetic sequences where each term increases by a constant difference.
• Geometric sequences where each term is found by multiplying the previous one by a constant.
• Sequences defined by a formula, such as \( \frac{1}{\sqrt{n}} \).

In the given exercise, the sequence is \( \frac{1}{\sqrt{1}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{4}} \), which follows the pattern \( \frac{1}{\sqrt{n}} \). Recognizing these patterns allows you to express the sequence in sigma notation for efficient summation.

Summation Index

The **Summation Index** is a critical element in sigma notation. It represents the variable that changes with each step in the summation process. Essentially, it is the variable that acts as a counter for the number of terms, determining the scope and extent of the summation. In the example of converting a series into sigma notation, understanding the summation index is important:

• The index is represented by a variable, usually \( n \), which steps through the numbers, indicating each term's position in the sum.
• The starting value of the index in our example is 1, and it increases incrementally until it reaches the end value, here 4.
• Thus, \( n \) varies from 1 to 4, covering all terms in the initial series.

By defining the summation index, you can determine the range of terms to be included in the summation. This understanding makes expressions neat and comprehensive, whether you're tackling evaluations or simplifying complex series.