Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate \(\sum_{k=0}^{\infty}(-1)^{k} x^{k},-1

Step-by-Step Solution

Answer

The sum of the series is \( \frac{1}{1 + x} \) for \(-1 < x < 1\).

Step 1: Identify the series type

The given series is \( \sum_{k=0}^{\infty} (-1)^k x^k \). This series is an alternating geometric series with first term \( a = 1 \) and common ratio \( r = -x \).

Step 2: Determine the convergence criteria

A geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if the absolute value of the common ratio \( |r| < 1 \). In this series, the common ratio \( r = -x \), so it converges when \( |-x| = |x| < 1 \). This is given in the problem statement as \(-1 < x < 1\).

Step 3: Apply the geometric series sum formula

The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. For this series, \( a = 1 \) and \( r = -x \), so the sum is \( S = \frac{1}{1 - (-x)} = \frac{1}{1 + x} \).

Step 4: Ensure conditions are met

Finally, verify that the conditions \(-1 < x < 1\) are met for the geometric series formula to be applied, which they are as stated in the problem.

Key Concepts

Series Convergence

Understanding series convergence is crucial when working with infinite series. Convergence essentially tells us whether the terms of a series approach a finite sum as more and more terms are added. In our context, we have an alternating geometric series where the terms alternate in sign and follow a geometric progression.

Series convergence depends on the common ratio of the geometric series. When the absolute value of the common ratio \(|r|\) is less than 1, the series will converge. This means we can sum the infinite terms to reach a specific value. For the alternating geometric series \( \sum_{k=0}^{\infty} (-1)^k x^k \), the common ratio \(r\) is \(-x\). As per the convergence criteria, \(|-x| = |x| < 1\) is the condition which ensures that our series converges for \( -1 < x < 1 \).

Convergence is confirmed by this range, allowing us to use the geometric series formula effectively to find the sum.

Geometric Series Formula

The geometric series formula is a powerful tool that simplifies computing the sum of infinite series when the series is geometric. A geometric series has a first term \(a\) and a common ratio \(r\), with each term being the product of the previous term and \(r\).

The sum of an infinite geometric series, which converges, can be calculated using the formula:

• \( S = \frac{a}{1 - r} \)

Here:

• \( S \) is the sum of the series
• \( a \) is the first term
• \( r \) is the common ratio

In our exercise, the series \( \sum_{k=0}^{\infty} (-1)^k x^k \) has \( a = 1 \) and \( r = -x \). Applying this formula gives us \( S = \frac{1}{1 + x} \) under the condition that \(-1 < x < 1\).

This formula helps to quickly evaluate series sums without having to add up infinitely many terms individually, provided the condition for convergence is met.

Infinite Series

Infinite series involve adding an infinite number of terms together. It might sound counterintuitive, but under certain conditions, you can actually reach a finite sum. This idea is the backbone of calculus and many areas of mathematical analysis.

Infinity can often be complicated, but when dealing with series like our geometric series, specific rules and formulas make it manageable. For example, if we know the series converges (meaning it approaches a particular value), we can confidently use formulas like the geometric series sum to find that value.

An infinite series does not automatically have a sum. The criteria for a series like \( \sum_{k=0}^{\infty} (-1)^k x^k \) relies heavily on the behavior of its terms. Proper understanding of conditions like \(|x| < 1\) ensures we can process the series with confidence. This insight allows mathematicians and students to deal with infinite concepts in practical scenarios efficiently.