Problem 66 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the General Term

First, observe the sequence given in the exercise: \(\frac{\sqrt{1}}{1^2}, \frac{\sqrt{2}}{2^2}, \frac{\sqrt{3}}{3^2}, \ldots, \frac{\sqrt{n}}{n^2}\). The general form of the sequence can be expressed as \(\frac{\sqrt{k}}{k^2}\), where \(k\) ranges over the positive integers from 1 to \(n\).

Step 2: Define the Sigma Notation Parameters

To express the sum in sigma notation, identify the variable that changes and its range. Here, the variable \(k\) changes from 1 to \(n\). This forms the limits of summation in sigma notation.

Step 3: Write the Sigma Notation

Combine the general term with the sigma notation parameters. The sum can be written using sigma notation as: \[\sum_{k=1}^{n} \frac{\sqrt{k}}{k^2}\] This represents the sum of all terms from \(\frac{\sqrt{1}}{1^2}\) to \(\frac{\sqrt{n}}{n^2}\).

Key Concepts

General Term

Understanding the "general term" is key to comprehending sequences and series. In mathematics, a general term is a formula that defines a sequence. Each term in the sequence can be generated using this formula by plugging in natural numbers like 1, 2, 3, and so on.
For the expression \( \frac{\sqrt{k}}{k^2} \), which represents this particular sequence, the variable \( k \) acts as a placeholder for these natural numbers. Using this general term, we systematically calculate each element of the sequence at different values of \( k \).
Each part of the formula has a specific role:

• The square root \( \sqrt{k} \) introduces a non-linear growth factor.
• The denominator \( k^2 \) ensures each term decreases in value as \( k \) increases.

This understanding aids in not just writing expressions but also analyzing how sequences behave as they progress.

Sequence

A sequence in mathematics is an ordered set of numbers, which follow a specific pattern or rule. In this context, each number or term is defined by a general formula. Sequences are an integral part of mathematical analysis and serve as building blocks for series and functions.
The sequence in the given problem is: \( \frac{\sqrt{1}}{1^2}, \frac{\sqrt{2}}{2^2}, \frac{\sqrt{3}}{3^2}, \ldots, \frac{\sqrt{n}}{n^2} \). To identify a sequence, you start by pinpointing the pattern or rule governing it.
The sequence follows the rule given by the general term \( \frac{\sqrt{k}}{k^2} \). This tells us that each subsequent term differs based on its position \( k \) in the sequence.
Recognizing sequences helps us predict future terms, analyze the trend, and later sum these parts if necessary.

Summation

Summation is the operation of adding a list of numbers together, often resulting in a series. Using sigma notation \( \Sigma \), mathematicians can compactly express the sum of a sequence's terms.
Sigma notation is not only concise but also elegantly links sequences to their summed result. The formula you analyze: \( \sum_{k=1}^{n} \frac{\sqrt{k}}{k^2} \), represents such a summation operation.
Here's how these parts work together:

• \( \Sigma \) indicates that a summation is to be performed.
• The index \( k=1 \) to \( n \) specifies the bounds. It means you start at the first term and proceed to the \( n \)th term.
• The expression \( \frac{\sqrt{k}}{k^2} \) is the general term we discussed, plugged into the sigma notation structure.

Understanding how to properly write and interpret summation notation is essential in calculating overall sums efficiently.