Problem 65 - Precalculus Mathematics for Calculus

Question

Write the sum using sigma notation. $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{999 \cdot 1000}$$

Step-by-Step Solution

Answer

\( \sum_{n=1}^{999} \frac{1}{n(n+1)} \)

Step 1: Identify the Pattern

First, observe the given series \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \cdots + \frac{1}{999 \cdot 1000} \). Each term has the form \( \frac{1}{n(n+1)} \), where \( n \) is a positive integer starting at 1.

Step 2: Determine the Range of Values

Determine the range of values for \( n \). The series starts at \( n = 1 \) and ends at \( n = 999 \), because the last term is \( \frac{1}{999 \cdot 1000} \).

Step 3: Write in Sigma Notation

Using the pattern identified, we can write the sum in sigma notation as:\[\sum_{n=1}^{999} \frac{1}{n(n+1)}\]This expression covers the entire sequence from \( n = 1 \) to \( n = 999 \), inclusive.

Key Concepts

Sequence and Series

In mathematics, sequences are ordered lists of numbers that follow a particular pattern, while series are the sum of terms of sequences. Understanding the difference between these two is crucial when working with sigma notation.

• A sequence is essentially an ordered list of numbers, such as 1, 2, 3, and so on. In our exercise, the sequence consists of the fractions \( \frac{1}{1\cdot 2}, \frac{1}{2\cdot 3}, \ldots, \frac{1}{999\cdot 1000} \).
• A series is the sum of the terms of a sequence. Here, we are tasked with finding the sum of the sequence of fractions, represented as a series.
• Sigma notation, also known as summation notation, is a way of expressing the sum of a sequence of terms. It provides a concise way of representing the series by noting the range and the general term structure.

The sequence quickly evolves into a related series when combined by addition. This transformation is where the beauty of patterns in mathematical sequences becomes evident.

Mathematical Patterns

Mathematics is full of fascinating patterns, one of which we recognize in the given problem. Identifying the pattern in a sequence or series is a key step toward simplifying complex mathematical expressions.
To solve the problem at hand, observe the mathematical pattern which is repeated across the series:

• Each element of the sequence fits the form \( \frac{1}{n(n+1)} \).
• Here, \( n \) represents a sequence of integers starting from 1 and progressing to 999.
• This consistent structure allows the use of algebraic techniques, such as partial fraction decomposition, to simplify or interpret the series.

Recognizing patterns simplifies understanding and solving problems, leading to a clearer representation of mathematical ideas. Moreover, this pattern assists in writing the terms concisely using sigma notation, covering all the terms uniformly.

Sum of Fractions

Fractions are one of the fundamental numerical forms in mathematics, and summing them requires a sound understanding of their structure. Our exercise deals with a sequence of fractions of the form \( \frac{1}{n(n+1)} \).

• This fraction resembles a telescoping series, opening opportunities for simplification.
• Given the structure, we can use the technique of partial fractions to break down the expression into components that can be easily summed.
• The process involves expressing the fraction as a difference: \( \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \). This representation significantly simplifies the summing of series because most terms cancel out.

In conclusion, when summed, you'll find that only the first and last terms make significant contributions because intermediate terms effectively "telescope" into each other. This crucial ability to break down a sum of fractions leads us closer to understanding and solving problems efficiently.