Problem 64 - Precalculus: Mathematical for Calculus

Question

Write the sum using sigma notation. $$\frac{\sqrt{1}}{1^{2}}+\frac{\sqrt{2}}{2^{2}}+\frac{\sqrt{3}}{3^{2}}+\cdots+\frac{\sqrt{n}}{n^{2}}$$

Step-by-Step Solution

Answer

The sum is \( \sum_{k=1}^{n} \frac{\sqrt{k}}{k^2} \).

Step 1: Understand the Series

We are given a series where each term takes the form \( \frac{\sqrt{k}}{k^2} \) and \( k \) ranges over integers starting from 1 up to \( n \). Our task is to write this series using sigma notation.

Step 2: Identify the General Term

Identify the general term of the series as \( \frac{\sqrt{k}}{k^2} \), where \( k \) is the index of summation and varies from 1 to \( n \). This allows us to express each term consistently.

Step 3: Set the Index of Summation

Determine the starting and ending values for \( k \), which are 1 and \( n \) respectively, since the series sums terms starting at \( k = 1 \) and ending at \( k = n \).

Step 4: Write Using Sigma Notation

Combine the general term and limits into the sigma notation: \[\sum_{k=1}^{n} \frac{\sqrt{k}}{k^2}\] This notation represents the sum of the terms from \( k=1 \) to \( n \) inclusive.

Key Concepts

Series Representation

In mathematics, a series represents the summation of individual terms of a sequence. A sequence is a list of numbers ordered in a particular manner. When we sum the terms of this sequence, we get a series. For example, the series you encounter when solving problems often takes on a specific pattern. The given series, \( \frac{\sqrt{1}}{1^{2}}+\frac{\sqrt{2}}{2^{2}}+\frac{\sqrt{3}}{3^{2}}+\cdots+\frac{\sqrt{n}}{n^{2}} \), follows a pattern where each term can be expressed generally as \( \frac{\sqrt{k}}{k^2} \). Here, \( k \) represents each step or term number in the sequence, ranging from 1 to \( n \). Understanding how to represent this series systematically makes complex calculations manageable. Series representation allows us to see the broader picture or pattern, especially as the number of terms \( n \) becomes large. This systematic approach is fundamental in calculus and other advanced math fields.

Summation

Summation is the process of adding a sequence of numbers. It is often used to combine elements systematically in a mathematical expression. The concept of summation becomes crucial when dealing with sequences and series. By recognizing that each element follows a specific rule or pattern, summation lets us compute the total in a concise way. In the example given, we use the general term \( \frac{\sqrt{k}}{k^2} \) for the darkened series. This tells us that each term is derived by applying this pattern to successive integers from 1 through \( n \).
Understanding summation simplifies many mathematical problems. By defining a start and end point, such as \( k = 1 \) to \( k = n \), we can compactly represent what would otherwise be a lengthy arithmetic process. The summation symbol \( \Sigma \) is a powerful tool that condenses this information into a more manageable form, offering a clear, visual representation of the sum of terms.

Mathematical Notation

Mathematical notation is a language that allows mathematicians to express ideas clearly and concisely. It uses symbols and expressions to convey complex concepts in a compact form. One crucial form of mathematical notation is sigma notation, represented by the Greek letter \( \Sigma \). This notation is particularly useful for depicting the sum of a series. When writing a series using sigma notation, the general term is paired with the index of summation, which indicates which terms in the sequence are being summed up. For the given exercise, sigma notation is shown as \[\sum_{k=1}^{n} \frac{\sqrt{k}}{k^2}\]. Here, \( k \) is the index of summation, starting from 1 and ending at \( n \), and \( \frac{\sqrt{k}}{k^2} \) is the formula for each term in the sequence.
Learning to interpret and write mathematical notation is essential for any student, as it allows them to engage with mathematical problems at a deeper level. Understanding the elegance of mathematical notation can transform how one sees and solves problems in mathematics.