Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Express the sum in terms of summation notation. (Answers are not unique.) $$\frac{1}{4}-\frac{1}{12}+\frac{1}{36}-\frac{1}{108}$$

Step-by-Step Solution

Answer

\( \sum_{n=1}^{4} (-1)^{n+1} \frac{1}{4 \cdot 3^{n-1}} \)

Step 1: Identify the Pattern

The given sequence is \( \frac{1}{4}, -\frac{1}{12}, \frac{1}{36}, -\frac{1}{108} \). Notice the alternating signs and the denominators which form a sequence: 4, 12, 36, 108.

Step 2: Notice the Denominator Pattern

Observe the denominators: 4, 12, 36, 108. These denominators can be expressed as powers of 3. Specifically, \( 4 = 3^0 \times 4, 12 = 3^1 \times 4, 36 = 3^2 \times 4, 108 = 3^3 \times 4 \). This can also be simplified as \( 4 \times 3^{n-1} \) where \( n = 1, 2, 3, 4 \).

Step 3: Recognize the Alternating Series

The numerators are 1 for each term and the series alternates signs. Thus, this can be expressed as \( (-1)^{n+1} \) to reflect the pattern of switching between positive and negative terms.

Step 4: Write the General Term and Summation

Combine these observations to write the general term for the series. The general term is \( (-1)^{n+1} \frac{1}{4 \cdot 3^{n-1}} \). Therefore, the series in summation notation is: \[ \sum_{n=1}^{4} (-1)^{n+1} \frac{1}{4 \cdot 3^{n-1}} \]

Step 5: Summarize the Expression

The final expression in summation notation is \( \sum_{n=1}^{4} (-1)^{n+1} \frac{1}{4 \cdot 3^{n-1}} \). This represents the original sequence provided in the exercise.

Key Concepts

Alternating Series

An alternating series is a sequence of numbers where the terms regularly switch between positive and negative. This is easy to spot because the sign of each successive term is different from the one before it. For example, in the series

• \( \frac{1}{4} \)
• \( -\frac{1}{12} \)
• \( \frac{1}{36} \)
• \( -\frac{1}{108} \)

the signs change from positive to negative as we move from one fraction to the next. These alternating signs are an essential characteristic of this type of series. We can represent an alternating series in summation notation using the term \((-1)^{n+1}\), which effectively flips the sign with each successive term in the sequence.

Geometric Sequence

A geometric sequence refers to a sequence where each term is derived by multiplying the previous term by a constant factor. In the exercise, the denominators

• 4
• 12
• 36
• 108

form a geometric sequence. Each denominator is obtained by multiplying the previous term by 3, which is the common ratio in this sequence. This characteristic defines the sequence as geometric. The general form of a geometric sequence can be written as \( a, ar, ar^2, ar^3, \ldots \) where \( a \) is the first term and \( r \) is the common ratio. In our series, the pattern \( 4 \times 3^{n-1} \) is recognizable, where the factor 4 can be thought of as a constant multiplier, making the denominators a part of a geometric sequence starting with 4.

Mathematical Series

A mathematical series is the summation of the terms of a sequence. Each individual term in the sequence is called an element, and when combined, they form a series. In the given exercise, the series combines several fractions with alternating signs. Using summation notation allows us to express a series compactly and clearly. In this case, the series is about summing terms of an alternating geometric sequence, expressed succinctly as:\[ \sum_{n=1}^{4} (-1)^{n+1} \frac{1}{4 \cdot 3^{n-1}} \]This concise form communicates the pattern and rules governing each term's place and value, as well as how they should be added together in sequence.

General Term

The general term of a sequence or series is a formula that represents any term in the sequence as a function of the position number \( n \). The ability to express the general term is crucial because it allows us to identify and create series using a formulaic approach, without listing each term individually. In the given exercise, the general term is indicated as:\[ (-1)^{n+1} \frac{1}{4 \cdot 3^{n-1}} \]This term incorporates several findings:

• The \((-1)^{n+1}\) alternates the sign based on whether \( n \) is odd or even.
• \( \frac{1}{4 \cdot 3^{n-1}} \) represents the \( n \)-th term of the geometric sequence in the denominators.

With this formula, we can generate each term of the series by simply substituting different values of \( n \). This systematic representation makes handling complex sequences straightforward and efficient.