Problem 16 - University Calculus: Early Transcendentals

Question

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$-\frac{1}{5}+\frac{2}{5}-\frac{3}{5}+\frac{4}{5}-\frac{5}{5}$$

Step-by-Step Solution

Answer

The sum is expressed in sigma notation as: \( \sum_{n=1}^{5} \frac{(-1)^n \, n}{5} \).

Step 1: Identify the Pattern

Observe the given sequence: \[-\frac{1}{5}, +\frac{2}{5}, -\frac{3}{5}, +\frac{4}{5}, -\frac{5}{5}\].It is an alternating sequence where each term has the form \( \frac{n}{5} \) with alternating signs.

Step 2: Determine the General Term

Identify the general term of the sequence. The numerators \(-1, 2, -3, 4, -5 \) form an arithmetic sequence with alternating signs, represented by \( (-1)^n \cdot n \). Consequently, the general form of each term can be written as \( \frac{(-1)^n \, n}{5} \).

Step 3: Decide on the Range and Index

From the sequence, it is observed that for \( n = 1, 2, 3, 4, 5 \), the sequence matches the given terms. So, we can use these values for the range of summation.

Step 4: Write in Sigma Notation

Using the information from the previous steps, express the sum in sigma notation:\[ \sum_{n=1}^{5} \frac{(-1)^n \, n}{5} \]

Key Concepts

Alternating Sequence

An alternating sequence is a mathematical pattern where the signs of the terms alternate between positive and negative. This kind of sequence is characterized by its regular positive-negative switching, which makes it easier to detect in a series of numbers or terms.

Common features of an alternating sequence include:

• The presence of signs such as "+" and "-" in a repetitive manner.
• The sequence often follows another mathematical pattern, such as an arithmetic or geometric series, on its magnitude or value.

In our exercise, the sequence alternates between negative and positive terms: o \(-\frac{1}{5}, +\frac{2}{5}, -\frac{3}{5}, +\frac{4}{5}, -\frac{5}{5}\). The alternation is specified by \((-1)^n\) in the sequence. This term changes the sign of each corresponding number in the pattern. Alternating sequences are frequently utilized in mathematics due to their predictable behavior.

General Term

The general term of a sequence is a formula or expression that allows us to find any term within the sequence. By using this formula, we don't have to reconstruct each term step-by-step from the beginning. We can simply input the position number—or index—of the term we want to find, and the formula calculates it.

To identify a general term, it's important to determine the factors that affect each term's value. In alternating sequences like ours, the general term incorporates both arithmetic features and the sign change:

\[(-1)^n \times \frac{n}{5}\] Here:

• \((-1)^n\) adjusts the sign alternation for each term.
• \(\frac{n}{5}\) accounts for the sequence's arithmetic increment, where \(n\) represents the term's position.

Unified within the general term, these components precisely define each term in the sequence for any position \(n\).

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the "common difference," and it's a defining trait of arithmetic sequences.

In our reading, the numerators o \(-1, 2, -3, 4, -5\) form an arithmetic sequence. Here, each consecutive number increases by a consistent value, specifically by adjusting for the alternating signs. Without the sign, the absolute values increase constantly by one.

• Common Difference: The pattern incrementally moves by 1 in its unsigned form.
• Role of \( (-1)^n \): This adds the alternating property to the sequence.

Arithmetic sequences underpin many mathematical constructs, serving as foundational elements that, when combined with other sequences like geometric or sinusoidal forms, show a wide variety of applications in mathematical theory and practical problem-solving.