Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The sum is \( \frac{1}{1+x} \).

Step 1: Recognize the Series

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

Step 2: Determine the Formula for a Geometric Series

The sum of an infinite geometric series with first term \(a\) and common ratio \(|r| < 1\) is given by \( \frac{a}{1 - r} \). This holds because the absolute value of the common ratio must be less than 1 for convergence.

Step 3: Substitute Values into the Formula

Substitute \(a = 1\) and \(r = -x\) into the formula. We have: \[ S = \frac{1}{1 - (-x)} = \frac{1}{1 + x} \].

Step 4: State the Condition for Convergence

Since the series is geometric, it converges under the condition \(|r| = |-x| < 1\), which simplifies to \(-1 < x < 1\). This condition is already provided in the problem statement, ensuring the series converges.

Key Concepts

Geometric Series

A geometric series is a special type of series in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This ratio remains constant throughout the series.
For example, in the series \( 1 + x + x^2 + x^3 + \ldots \), each term is acquired by multiplying the previous term by \( x \). Recognizing that a series is geometric helps in identifying its formula for summation.

• The sum of an infinite geometric series is expressed as:
  \( S = \frac{a}{1-r} \)
• Here, \( a \) represents the first term, and \( r \) denotes the common ratio.
• It is critical that the absolute value of the common ratio \( |r| \) be less than 1 for the series to converge.

This formula arises from observing that as we sum an increasingly large number of terms, the terms’ contributions diminish, converging towards a predictable limit.

Series Convergence

Series convergence refers to the condition wherein the sum of the terms of an infinite series approaches a finite number as the number of terms increases indefinitely. Understanding convergence helps determine whether a series sums to a limit or diverges.
Factors influencing convergence include the nature of the series and the magnitude of its terms. For instance:

• An infinite series like the geometric series converges if and only if the absolute value of the common ratio is less than one.
• Convergence implies that, in practical terms, adding more terms after a point does not significantly change the sum.

For the geometric series presented in the exercise, the expression \( |r| < 1 \) directly dictates convergence, ensuring the series has a sum.

Infinite Series

An infinite series, as its name suggests, consists of infinitely many terms, and the sum is taken over all these terms. These series are represented with the sigma notation \( \sum \). Infinite series can sometimes add up to a finite value when a clear pattern is identified, particularly in geometric series.
Key points regarding infinite series include:

• In the context of geometric series, the condition \( |r| < 1 \) ensures that the sum converges to a limit rather than diverging to infinity.
• While the idea of adding infinitely many numbers might seem counterintuitive, it's a fundamental concept in calculus because it allows us to approach the limit of sums rigorously.
• Evaluating an infinite series often involves finding such a limit, where the sum of an infinite number of terms becomes manageable and finite under certain conditions.

This approach turns the theoretical idea of infinite addition into practical calculations used in various scientific and engineering fields.