Problem 126 - 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

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

Step 1: Identify the Pattern of the Terms

We begin by examining the sequence of terms in the sum: \(\frac{1}{\sqrt{1}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{4}}\). Notice that each term can be expressed as \(\frac{1}{\sqrt{k}}\) where \(k\) is a positive integer.

Step 2: Determine the Range of the Index

Next, we determine the range of \(k\) for this sequence. The terms start at \(\frac{1}{\sqrt{1}}\) and end at \(\frac{1}{\sqrt{4}}\). Therefore, the index \(k\) starts at 1 and ends at 4.

Step 3: Write the Sigma Notation

Now, combine the pattern and the range into sigma notation. The sum can be written as:\[\sum_{k=1}^{4} \frac{1}{\sqrt{k}}\] This notation means that you sum the expression \(\frac{1}{\sqrt{k}}\) as \(k\) ranges from 1 to 4.

Key Concepts

Sequence of Terms

When you first look at a sum like \( \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} \), it's all about recognizing how each component fits into a larger scheme. First, let's break down the concept of "sequence of terms." A sequence is simply an ordered list of numbers. It's like a number train where each car is hitched by a rule or pattern. For the sum we're dealing with, the sequence is formed by each term being one fraction with a square root in the denominator: \( \frac{1}{\sqrt{k}} \). The variable \( k \) acts as a placeholder that tells you which position a term holds in the sequence.

• In this problem, the first term is when \( k = 1 \), leading to \( \frac{1}{\sqrt{1}} \).
• The second term is \( \frac{1}{\sqrt{2}} \) when \( k = 2 \).
• This pattern continues until the final term \( \frac{1}{\sqrt{4}} \) for \( k = 4 \).

So, the sequence is driven by a simple mathematical pattern, governed here by \( k \) ranging from 1 to 4.

Sum in Calculus

The idea of a "sum" in calculus is an extension of basic arithmetic sums but with added depth. Calculus often studies sums because they help in understanding rates of change and can be the building blocks to more complex ideas like integration.
In this example, even though it's simply adding fractions, that sum tells you much about how values behave as you move through the sequence. Sigma notation \(\sum\) is the language calculus uses to compactly express sums with many terms, especially when a clear pattern is present.By writing \(\sum_{k=1}^{4} \frac{1}{\sqrt{k}}\), it's understood that there is:

• A fixed beginning and end for \( k \) (in our case, from 1 to 4).
• A repeated action of adding each term within that range according to the set pattern \( \frac{1}{\sqrt{k}} \).

Although this sum doesn't employ other calculus tools like derivatives or integrals, understanding sigma notation is a critical foundation for later calculus topics.

Pattern Identification

Recognizing a pattern is crucial both for writing sums in sigma notation and for solving many mathematical problems. A pattern gives you a way to predict and validate the elements of a sequence.The first step in solving the original problem was to identify the pattern that defines each term in the sum.

• Here, each term in the sequence could be described as \( \frac{1}{\sqrt{k}} \).
• By identifying this formula or rule early on, you clarify what connects one term to the next and ensures all the calculations are consistent.
• This understanding makes the transition to sigma notation straightforward, as the pattern itself becomes the expression inside the sum.

Pattern identification isn't just about finding a formula—it also involves recognizing the characteristics of the terms such as their arithmetic or geometric relationships. In our case, the key was seeing the term so-wrapped with square roots and fractions, then simplifying it to the concise expression used in the sum.