Problem 56 - Calculus Early Transcendentals

Question

Write the following power series in summation (sigma) notation. $$1+\frac{x}{2}+\frac{x^{2}}{4}+\frac{x^{3}}{6}+\cdots$$

Step-by-Step Solution

Answer

Question: Represent the infinite power series as a sigma notation: $$\frac{1}{2} + \frac{x}{4} + \frac{x^2}{6} + \frac{x^3}{8} + \cdots$$ Answer: The infinite power series can be represented using sigma notation as: $$\sum_{n=1}^{\infty} \frac{x^{n-1}}{2n}$$

Step 1: Determine the general nth term of the series

Observe the pattern in the terms of the series. Notice that for each term, the exponent of x is equal to n - 1, where n is the position of the term in the series (starting from 1). Moreover, the denominator of each term is equal to n times 2. Hence, the general term can be written as: $$a_n = \frac{x^{n-1}}{2n}$$

Step 2: Write the series in sigma notation

Now that we have the general term, we can represent this series using sigma notation and let n start from 1 and go to infinity: $$\sum_{n=1}^{\infty} \frac{x^{n-1}}{2n}$$

Key Concepts

Summation Notation

Summation notation is a convenient and compact way to represent the sum of a sequence of numbers. Instead of writing out each term of the series, we use a concise form involving a summation symbol (which looks like an "E" on its side). This symbol tells us to add up everything that follows.For example, the series involving terms like 1, \( \frac{x}{2} \), and \( \frac{x^{2}}{4} \) can be described without listing every element. Instead, we say it is a summation of terms following a particular pattern. Here, we notice that each term involves powers of x and decreasing fractions.This format saves space and makes complex series easier to manage, especially when dealing with many terms or even infinite series.

Sigma Notation

Sigma notation is defined using the Greek letter \( \Sigma \), representing the sum. It captures the pattern of the terms in a series. In our power series, each term follows a predictable rule or formula. This formula is what we represent within the sigma notation.To use sigma notation:

• Identify the general term of the series.
• Write the sigma symbol: \( \Sigma \).
• Substitute the general term formula inside the sigma notation.
• Determine the range of the sum, such as from \( n = 1 \) to infinity.

By using sigma notation, we transform the series into a more manageable and universally understandable form. Thus, the series \( 1 + \frac{x}{2} + \frac{x^{2}}{4} + \dots \) is written as \( \sum_{n=1}^{\infty} \frac{x^{n-1}}{2n} \).

General Term

The general term is like the secret recipe for the whole series. It tells you exactly how to create each term based on its position or index in the sequence. For our series, each term's numerator is formed by raising \( x \) to the power of the position minus one, such as \( x^{n-1} \).The denominator is a bit different and involves multiplying 2 by the position number, \( n \). This gives us a complete formula for each term in the series: \( a_n = \frac{x^{n-1}}{2n} \).Knowing the general term:

• The exponent of \( x \) increases by one each step.
• The denominator increases proportionately with the position number.

This formula allows us to express any term in the series efficiently.

Infinite Series

An infinite series is like a series of numbers that never stops. It keeps on going forever, adding each subsequent term. Our series is an infinite series because it doesn't end at a specific point; instead, it continues endlessly.In mathematics, writing an infinite series in summation or sigma notation often includes the upper limit of infinity, denoted by \( \infty \). This tells you that there are infinitely many terms to be summed up, such as in \( \sum_{n=1}^{\infty} \frac{x^{n-1}}{2n} \).Thinking about infinite series involves:

• Recognizing the series goes on without end, never reaching a final sum.
• Understanding the concept of convergence, where the sum comes very close to a fixed value as more terms are added.

While infinite series may seem daunting, they're important in calculus and advancing mathematical concepts.