Problem 72 - Precalculus: Mathematics 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

\( \sum_{k=1}^{n} \frac{\sqrt{k}}{k^{2}} \)

Step 1: Identify the General Term

The first step is to identify the general structure of the terms given in the sequence: \( \frac{\sqrt{1}}{1^{2}}, \frac{\sqrt{2}}{2^{2}}, \frac{\sqrt{3}}{3^{2}}, \cdots, \frac{\sqrt{n}}{n^{2}} \). For the \(k\)-th term, it is represented by \( \frac{\sqrt{k}}{k^{2}} \).

Step 2: Recognize the Pattern Limits

Determine the limits of the sum. The given expression starts from \(k=1\) and proceeds to \(k=n\).

Step 3: Write in Sigma Notation

Now convert the series into sigma notation. The general form of the terms \( \frac{\sqrt{k}}{k^{2}} \) and the determined limits of \(k=1\) to \(k=n\) allow us to write \[\sum_{k=1}^{n} \frac{\sqrt{k}}{k^{2}}\] as the sigma notation for the given series.

Key Concepts

Sequence and Series

Sequences and series are fundamental concepts in precalculus mathematics. A **sequence** is an ordered list of numbers following a particular pattern, where each number is called a term. For instance, in the context of the given exercise, the sequence consists of terms like \(\frac{\sqrt{1}}{1^2}, \frac{\sqrt{2}}{2^2}, \frac{\sqrt{3}}{3^2}, \ldots\). Each term follows a specific rule based on its position in the sequence, often expressed as a function of an integer variable such as \(k\).

A **series** is the sum of the terms of a sequence. When you add these terms together, starting from the first term to the \(n\)-th term, you get a series. In the given problem, the series is the sum of the sequence up to the \(n\)-th term. Understanding sequences and series is crucial as they form the backbone of many mathematical concepts and are stepping stones to more advanced topics like calculus.

Precalculus Mathematics

Precalculus mathematics includes various topics such as algebra, trigonometry, and the study of sequences and series, which prepare students for calculus. It builds a solid foundation by introducing concepts that help in understanding the behavior of functions and their limits. Through precalculus, students learn about how functions behave, and how they can be manipulated and analyzed to solve problems.

In the context of the problem, precalculus helps in recognizing patterns in sequences and transforming them into series. It also teaches important skills, such as identifying general terms in sequences and using notation systems such as sigma notation to express these series compactly and clearly. Mastering these skills is essential for successfully transitioning into calculus, where the concept of limits takes on a central role.

Summation Notation

Summation notation, also known as sigma notation, provides a concise way to represent the sum of a sequence's terms. Using the Greek capital letter \( \Sigma \), this notation simplifies the expression of lengthy sums by indicating both the form of the terms and the range of indices over which to sum them.

In the original exercise, the sum is represented as \(\sum_{k=1}^{n} \frac{\sqrt{k}}{k^2}\). Here, \(\Sigma\) conveys that we are summing terms starting from \(k=1\) up to \(k=n\). The expression \(\frac{\sqrt{k}}{k^2}\) tells us the form of each term in the sequence.

Summation notation is both a tool of simplification and one of clarity. It allows mathematicians and students to handle large series efficiently, enabling the focus on the properties and applications of the series, rather than their often cumbersome expanded forms. Understanding this notation is a fundamental part of mathematical literacy in precalculus and beyond.