Problem 71 - 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 General Pattern

Each term in the series follows the pattern \( \frac{1}{n(n+1)} \). Here, the sequence of natural numbers begins at 1.

Step 2: Define the Index Range for the Sigma Notation

The given series starts with \( n = 1 \) and ends with \( n = 999 \), according to the final term \( \frac{1}{999 imes 1000} \).

Step 3: Express the Series Using Sigma Notation

Utilize the general term for the series and the index range to write the summation in sigma notation: \( \sum_{n=1}^{999} \frac{1}{n(n+1)} \).

Key Concepts

Series

In mathematics, a series is the sum of the terms of a sequence. Imagine you have a sequence, which is just a list of numbers arranged in a specific order. When you decide to add these numbers together, you've formed a series. The list of numbers in a sequence can form a specific pattern or rule.
For instance, in the exercise provided, each term in the series has the form \( \frac{1}{n(n+1)} \). This shows a logical pattern from one term to the next, where each denominator is the product of two consecutive integers.

• The concept of a series helps in finding quick sums for large sets of numbers.
• Series can be finite, like our example that ends at 999, or infinite, continuing indefinitely.

Series are fundamental in many areas of mathematics, including calculus, where you're often asked to find the sum of all elements of a sequence.

Summation

Summation is the process of adding a sequence of numbers; the result is a sum. To simplify this process, especially when dealing with lots of numbers or complex expressions, mathematicians use special notations, like the sigma notation.
In our exercise, the summation is indicated by a large sigma symbol (\( \Sigma \)), which is a fancy way of saying 'add up all of these things'. In the sigma notation \( \sum_{n=1}^{999} \frac{1}{n(n+1)} \), each part has a specific role:

• The \( \Sigma \) symbol tells us we're summing things up.
• \( n=1 \) to \( 999 \) specifies the starting and ending point of our summation, defining how many terms are involved.
• The expression \( \frac{1}{n(n+1)} \) represents what exactly we're adding.

This notation makes it easier to express the calculation of large sums without writing out all terms individually.

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers, operations, and concepts. It's essentially the language of mathematics, allowing complex ideas and calculations to be communicated succinctly. For our specific case, sigma notation is a form of mathematical notation used to express summation across multiple terms.
Understanding this notation allows us to work efficiently and avoid cumbersome expressions. In the exercise, sigma notation helps capture the essence of the problem without listing each term separately. It's especially helpful for lengthy sequences, where listing all terms would be impractical.

• Sigma notation simplifies the representation of a series.
• Helps focus on the pattern or formula of the terms, which is crucial for solving or simplifying complex expressions.
• Widely used in calculus, algebra, and number theory, making it an invaluable tool for students and mathematicians alike.

Knowing this notation is essential for progressing in more advanced mathematics and disciplines that heavily rely on mathematical analysis.