Problem 58 - Calculus Early Transcendentals

Question

Write the following power series in summation (sigma) notation. $$x-\frac{x^{3}}{4}+\frac{x^{5}}{9}-\frac{x^{7}}{16}+\cdots$$

Step-by-Step Solution

Answer

Question: Rewrite the given power series in summation (sigma) notation: $$x - \frac{x^3}{4} + \frac{x^5}{9} - \frac{x^7}{16} + \cdots$$ Answer: The given power series can be written in summation notation as follows: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{n^2}$$

Step 1: Identify the pattern in the term exponents

The exponents of the terms in the given power series are odd numbers. They follow the sequence 1, 3, 5, 7, and so on. We can represent this pattern using the formula: exponent = 2n-1 where n is the term number.

Step 2: Identify the pattern in the term denominators

The denominators of the terms in the given power series are squares of the corresponding term number. They follow the sequence 1, 4, 9, 16, and so on. We can represent this pattern using the formula: denominator = n^2 where n is the term number.

Step 3: Identify the sign of the terms

The terms of the given power series alternate between positive and negative. To represent this sign pattern, we will use the expression: \((-1)^{n+1}\), which will give a positive sign for odd n values and a negative sign for even n values.

Step 4: Write the given power series in summation notation (sigma notation)

Combine the patterns identified in steps 1-3 to rewrite the given power series in summation notation: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{n^2}$$

Key Concepts

Sigma Notation

Sigma Notation is a convenient way to write the summation of a sequence of terms. The symbol used, \(\Sigma\), is the Greek letter "sigma" which represents the word "sum." By using this notation, we can simplify complex series expressions into a more manageable form. When writing in sigma notation, a typical layout is:

• \(\Sigma\) symbol to indicate the sum
• A general term that describes each element in the sequence
• Limits of summation that define the starting and ending index of the series

For the given power series \( x - \frac{x^{3}}{4} + \frac{x^{5}}{9} - \frac{x^{7}}{16} + \cdots \), the pattern can be described as \( \sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^{2n-1}}{n^2} \). This shows the power series as a sum from \(n=1\) to infinity, with each term following the specified pattern.

Alternating Series

An Alternating Series is a series in which the terms alternate in sign. Essentially, the terms switch between positive and negative. This type of series can be expressed with the formula \((-1)^{n+1}\cdot a_n\) or \((-1)^n\cdot a_n\), depending on whether the series starts with a positive or negative term, respectively.

In our power series example \(\sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^{2n-1}}{n^2}\), the alternating sign is captured by \((-1)^{n+1}\). This ensures that the first term is positive and the sign alternates with each subsequent term, producing a series of the required pattern:

• Positive: when \(n = 1, 3, 5, \ldots\)
• Negative: when \(n = 2, 4, 6, \ldots\)

The alternating series is useful when calculating approximations or finding convergence of series.

Series Convergence

Series Convergence refers to the behavior of a series as the number of terms approaches infinity. A series is said to converge if its sequence of partial sums approaches a specific limit. When a series converges, adding more terms will not significantly change the sum.

• Divergent: A series is divergent if the sum does not approach any finite value.
• Convergent: A series converges if it approaches a finite sum as terms are added.

Alternating series, like our example, have specific criteria for convergence. The Alternating Series Test states that if the absolute value of terms \(a_n\) decreases monotonically (always getting smaller) and \(\lim_{n \to \infty} a_n = 0\), then the series converges.

This is particularly helpful for determining the usability of the power series in practical applications where convergence ensures the accuracy and stability of results.

Mathematical Patterns

Recognizing Mathematical Patterns is crucial when handling power series or any mathematical sequence. Patterns can significantly simplify the description of a sequence and make it easier to write in a closed form or specific notation like sigma notation.

• Exponent Pattern: In our series, the exponents of the variable \(x\) follow the odd number sequence \(1, 3, 5, 7, \ldots\).
• Denominator Pattern: The sequence \(1, 4, 9, 16, \ldots\) indicates the square numbers \(n^2\).
• Sign Pattern: The alternating sequence of signs, captured by \((-1)^{n+1}\).

Identifying these patterns enabled us to rewrite the series \( x - \frac{x^{3}}{4} + \frac{x^{5}}{9} - \frac{x^{7}}{16} + \ldots \) succinctly in sigma notation as \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{2n-1}}{n^2} \). Recognizing patterns can make working with series more intuitive and efficient.