Problem 21 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each series in sigma notation. $$ \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7}+\frac{7}{8}+\frac{8}{9}+\frac{9}{10} $$

Step-by-Step Solution

Answer

The series is written in sigma notation as \( \sum_{n=1}^{9} \frac{n}{n+1} \).

Step 1: Observe the Pattern

Examine the given series: \( \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7}+\frac{7}{8}+\frac{8}{9}+\frac{9}{10} \). Notice that each term follows the format \( \frac{n}{n+1} \) where \( n \) starts at 1 and increases by 1 for each subsequent term.

Step 2: Determine the Start and End Values

From the series, observe that the numerator of the first term is 1 and the numerator of the last term is 9. This tells us that \( n \) starts at 1 and ends at 9.

Step 3: Write the Series in Sigma Notation

Using the identified pattern and range of \( n \), write the given series in sigma notation as \( \sum_{n=1}^{9} \frac{n}{n+1} \).

Key Concepts

Series

A series is the sum of the terms of a sequence. In mathematics, when we add up the numbers in a pattern or list, it becomes what we call a "series." For example, in our exercise, we are summing the terms like \( \frac{1}{2}, \frac{2}{3}, \ldots, \frac{9}{10} \). This group of terms forms a series when summed up.
However, a series is not just any sum; it comes from something specific called a "sequence." The beauty of a series is that it allows us to express long sums concisely using mathematical notation.

In our exercise, we recognized that our series was formed by terms of the form \( \frac{n}{n+1} \). Using sigma notation, it helps us represent this efficiently from \( n = 1 \) to \( n = 9 \). Thus, the entire series is represented compactly as \( \sum_{n=1}^{9} \frac{n}{n+1} \).
The sigma (\( \Sigma \)) symbol plays a crucial role here, making it easier to handle such sums, whether by hand or computational methods.

Sequence

A sequence is a list of numbers arranged in a specific order. Each number in the list is called a term. Sequences are foundational in understanding series because they show us what's being added up. In the problem's context, each term follows a distinct pattern, \( \frac{n}{n+1} \), which simplifies spotting the logical pattern.

Let's break this down further:

• The sequence here starts by setting \( n = 1 \), so \( \frac{n}{n+1} = \frac{1}{2} \).
• Next, replace \( n = 2 \), so \( \frac{n}{n+1} = \frac{2}{3} \).
• Continue this until \( n = 9 \).
• The terms are all connected by the variable \( n \), each incrementing by 1 up to 9.

If you grasp what is happening in the sequence, you can better understand the series. Recognizing the pattern helps in writing or interpreting the series in sigma notation.

Algebra

Algebra is the branch of mathematics where we deal with symbols and the rules for manipulating those symbols. It is a key tool in representing patterns, especially in sequences and series, like the one in the problem.
The pattern \( \frac{n}{n+1} \) is an algebraic expression. It's essential to identify this expression to convert a series into sigma notation which then uses this algebraic formula, iterated across specified bounds.

Through algebra, we:

• Identify the relationship between terms (\( n = 1 \) to \( n = 9 \)).
• Use variables (e.g., \( n \)) to represent numbers and operations compactly.
• Simplify complex expressions or series into manageable algebraic forms.

By mastering the algebra behind sequences and series, we can handle even more significant mathematical problems efficiently.

Mathematical Notation

Mathematical notation is the language through which we communicate mathematical ideas precisely and efficiently. In our problem, it helps us write the series in a concise manner using sigma (\( \Sigma \)) notation.

Here's how it works:

• The sigma symbol (\( \Sigma \)) is used to indicate a summation. In our case, \( \sum_{n=1}^{9} \).
• Subscripts tell us where to start: \( n=1 \).
• Superscripts show where to end: \( n=9 \).
• The algebraic term \( \frac{n}{n+1} \) defines what we're summing.

By using mathematical notation, we turn a lengthy written sum into a compact, elegant expression. This not only saves time but also facilitates computations in more advanced mathematics or computer algorithms.