Problem 66 - Precalculus: Mathematics for Calculus

Question

Write the sum without using sigma notation. $$\sum_{j=1}^{n}(-1)^{j+1} x^{j}$$

Step-by-Step Solution

Answer

The sum is \( x - x^2 + x^3 - \ldots + (-1)^{n+1}x^n \).

Step 1: Understand the Sigma Notation

The sigma notation \( \sum_{j=1}^{n}(-1)^{j+1} x^{j} \) represents a sum where \( j \) starts at 1 and ends at \( n \). Each term in the sum has the form \((-1)^{j+1} x^{j}\).

Step 2: Interpret the Alternating Series Pattern

Notice that \((-1)^{j+1}\) creates an alternating pattern of signs: positive for even \( j \) and negative for odd \( j \).

Step 3: Expand the Series for Specific Terms

When \( j=1 \), the term is \( (-1)^{2}x^1 = x \), when \( j=2 \), the term is \( (-1)^{3}x^2 = -x^2 \). Keep alternating signs and increasing the power of \( x \) by 1 for each increment in \( j \): \( x - x^2 + x^3 - x^4 + \ldots \).

Step 4: Express the Entire Sum without Sigma Notation

Following the pattern obtained in Step 3, write the sum up to the nth term: \( x - x^2 + x^3 - x^4 + \ldots + (-1)^{n+1}x^n \).

Key Concepts

Sigma Notation

Sigma notation is a concise way to express a long sum of terms. It is especially helpful for identifying the structure of a series without writing each term individually. In the problem given, the sigma notation \( \sum_{j=1}^{n}(-1)^{j+1} x^{j} \) indicates a sum where each term is formed by plugging in successive integer values for \( j \) from 1 to \( n \). This notation clearly shows the pattern of each term involved in the series.

To break it down further:

• The index \( j \) in \( \sum \) tells us the position of the term in the series.
• The expression \( (-1)^{j+1} \) changes sign with each subsequent term.
• The term \( x^{j} \) represents the power of \( x \) that corresponds to the index \( j \).

Using sigma notation simplifies the representation of the sum, allowing quick insights into its structure. It also provides a foundation for further calculations and insights into the behavior of the series as \( n \) increases.

Series Expansion

Series expansion involves writing out the sum of terms expressed using sigma notation in their complete form. It's like unfolding a compact idea into its full, step-by-step detail. In the exercise provided, we see a transformation from sigma notation into an expanded series. This switch often helps in visualizing and understanding the underlying patterns of the series.

Starting with \( j=1 \), the term is \((-1)^{1+1} x^{1} = x\). For \( j=2 \), it becomes \((-1)^{2+1} x^{2} = -x^2\). This alternating sign pattern continues, yielding the visible sequence:

• \( x - x^2 + x^3 - x^4 + \ldots \)
• Each term's power increases sequentially (1, 2, 3, ...).
• The sign alternates between positive and negative.

By expanding the series, one clearly sees how the sum behaves, and this can be particularly useful in manually gauging convergence or divergence tendencies when considering more complex series.

Mathematical Patterns

Identifying mathematical patterns within a series is crucial for grasping its essence and predicting its behavior. The sequence in this exercise, \( x - x^2 + x^3 - x^4 + \ldots \), exemplifies such a pattern.

Understanding the aspects of this pattern helps in simplifying computations and predicting terms without the need to expand fully each time. Some notable features of the pattern include:

• Alternating Signs: Caused by the term \((-1)^{j+1}\), this results in the terms switching from positive to negative.
• Sequential Powers: The powers of \( x \) rise steadily from 1 onward, mirroring the index \( j \).
• Predictability: Once the pattern is noticed, predicting future terms becomes straightforward.

Grasping these patterns not only eases calculations but also sharpens overall understanding, offering an intuitive way to perform series manipulations and contemplate broader mathematical themes.