Problem 8 - Calculus Early Transcendentals

Question

How are the radii of convergence of the power series \(\sum c_{k} x^{k}\) and \(\sum(-1)^{k} c_{k} x^{k}\) related?

Step-by-Step Solution

Answer

Answer: Yes, the radii of convergence for both power series are the same.

Step 1: Examine the given power series

We have two power series to investigate: \(\sum c_{k} x^{k}\) and \(\sum(-1)^{k} c_{k} x^{k}\).

Step 2: Apply the Ratio Test

The Ratio Test is used to determine the convergence of a power series and is defined as: \(\lim_{k\to\infty} \left|\frac{c_{k+1} x^{k+1}}{c_{k} x^{k}}\right|\). For the given power series \(\sum c_{k} x^{k}\), let's apply the Ratio Test: $$\lim_{k\to\infty} \left|\frac{c_{k+1} x^{k+1}}{c_{k} x^{k}}\right| = \lim_{k\to\infty} |x|\left|\frac{c_{k+1}}{c_k}\right|$$ For the given power series \(\sum(-1)^{k} c_{k} x^{k}\), let's apply the Ratio Test: $$\lim_{k\to\infty} \left|\frac{(-1)^{k+1}c_{k+1} x^{k+1}}{{(-1)^k}c_{k} x^{k}}\right| = \lim_{k\to\infty} |x|\left|\frac{c_{k+1}}{c_k}\right|$$ Note that the negative sign (-1) to the power of k gets canceled out when we find the absolute value of the fractions.

Step 3: Compare the Radii of Convergence

From Step 2, we see that in both cases, the limit expressions, except for the constants \(c_k\), are the same: \(|x|\left|\frac{c_{k+1}}{c_k}\right|\). Thus, the radii of convergence for the power series \(\sum c_{k} x^{k}\) and \(\sum(-1)^{k} c_{k} x^{k}\) are the same.

Key Concepts

Radii of Convergence

In mathematics, particularly in the study of power series, the radius of convergence is a crucial concept. It determines the interval within which a power series converges. Take a power series of the form

• \( \sum c_{k} x^{k} \) where \( c_k \) are coefficients.
• The radius of convergence \( R \) tells us the values of \( x \) for which the series converges.

For example, if the radius of convergence is 3, it means the power series will converge for all \( x \) that are inside the interval \(-3 < x < 3\). When \( x \) exceeds these bounds, the series may diverge. Understanding radii helps to figure out how far a series 'stretches' before it stops giving meaningful results.
The radii of convergence of two series, like \( \sum c_{k} x^{k} \) and \( \sum(-1)^{k} c_{k} x^{k} \), are determined similarly. They depend on the same coefficient sequence \( c_k \), which is why their radii often align. This is the conclusion drawn from the ratio test regarding the radii of convergence of such series.

Ratio Test

The ratio test is a method to determine the convergence of an infinite series. It evaluates the behavior of terms in the series as they progress towards infinity. Here's how it works for a series \( \sum a_k \)

• Calculate \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
• If the limit is less than 1, the series converges absolutely.
• If it's more than 1, the series diverges.
• If it equals 1, the test is inconclusive.

When applying the ratio test to power series like \( \sum c_{k} x^{k} \) or \( \sum (-1)^{k} c_{k} x^{k} \), find the limit \[ \lim_{k \to \infty} |x|\left|\frac{c_{k+1}}{c_k}\right| \]This helps identify the radius of convergence by finding values of \( x \) where this limit is less than 1.
Noticeably, for both power series mentioned, even with an alternating sign, the ratio test's limit leads to the same expression for the radius of convergence.

Alternating Series

Alternating series have terms that switch between positive and negative. One common form is

• \( \sum (-1)^k a_k \).
• Here, every other term changes sign due to \((-1)^k\).

An alternating series can converge even if its non-alternating counterpart doesn’t. It depends on interval shrinking as terms progress. An important point is musing the absolute value for convergence tests.
When evaluating radii of convergence using the ratio test, we strip away the signs by taking absolute values. Hence, a series like \( \sum (-1)^{k} c_{k} x^{k} \) will have the same convergence criteria as \( \sum c_{k} x^{k} \). This feature highlights that while the signs alternate, they don’t affect the radius, revealing that the central convergence properties are preserved.