Problem 73 - College Algebra with Modeling and Visualization

Question

Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\frac{1}{5^{2}}+\cdots$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern in the Series

Observe that the series has terms of the form \( 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots \). Notice that the series consists of fractions where the numerators remain 1 and the denominators are squares of consecutive natural numbers starting from 1. This pattern suggests a general term: \( a_n = \frac{1}{n^2} \).

Step 2: Write the General Term

With the pattern identified, express the general term of the series. The general term is \( a_n = \frac{1}{n^2} \) for \( n = 1, 2, 3, \ldots \). This pattern can be used to express the series in summation notation.

Step 3: Express the Series with Summation Notation

Using the general term \( a_n = \frac{1}{n^2} \), write the series as an infinite sum from \( n = 1 \) to infinity. The series can be expressed as \( \sum_{n=1}^{\infty} \frac{1}{n^2} \). This notation captures the entire series as it continuously follows the identified pattern indefinitely.

Key Concepts

Series Representation

Series representation is a fundamental concept in mathematics that helps us depict lengthy or even infinite sequences concisely. Instead of listing every single term of the sequence, we use summation notation to represent the entire series in a compact form.

This exercise aims to represent the given series using summation notation. The series initially listed is

• 1
• \( \frac{1}{2^{2}} \)
• \( \frac{1}{3^{2}} \)
• \( \frac{1}{4^{2}} \)
• and so on...

The pattern of these terms helps us convert them into a simplified mathematical expression. By using the general term of the sequence in the given format, the entire sequence can be succinctly expressed as: \[\sum_{n=1}^{\infty} \frac{1}{n^2}\] With summation notation, we capture the essence of the series without writing each term repeatedly, showing both simplicity and efficiency.

General Term Identification

Identifying the general term in a series is crucial for transforming it into summation notation. This process involves recognizing a consistent pattern that the sequence follows.

In the given exercise, the series starts as 1, \( \frac{1}{2^2} \), \( \frac{1}{3^2} \), and so forth. The key observation here is:

• The numerators remain constant at 1.
• The denominators are successive squares of natural numbers, starting from 1.

This clear pattern allows us to formulate a general term for the series as \( a_n = \frac{1}{n^2} \).
For any term in the series, all we need to change is the value of \( n \) in \( \frac{1}{n^2} \), where \( n \) takes on natural number values such as 1, 2, 3, etc. Thus, identifying the general term provides a formulaic approach to the entire sequence, enabling conversion to summation notation effectively.

Pattern Recognition

Pattern recognition is a vital skill in determining sequences and series, especially when transforming them into more complex notations. It involves spotting regular, repeatable elements within the terms of a sequence.

The exercise presents a sequence: 1, \( \frac{1}{2^2} \), \( \frac{1}{3^2} \), ... up to infinity. Recognizing that the numerators are 1 and the denominators are squares of natural numbers allows the detection of an underlying pattern:

• The series starts from 1 and follows the form \( \frac{1}{n^2} \).
• The power of 2 in the denominator is consistent across terms.

By observing this structure, it becomes easier to determine that the series follows a simple pattern regarding its construction. The recognition pattern leads to the formulation of the general term \( a_n = \frac{1}{n^2} \), helping to express the entire series in neat and concise summation notation. Mastery of pattern recognition simplifies the process of interpreting, organizing, and transforming mathematical sequences.