Problem 29 - Calculus: Early Transcendentals

Question

If \( \sum_{n = 0}^{\infty} c_n4^n \) is convergent, can we conclude that each of the following series is convergent? (a) \( \sum_{n = 0}^{\infty} c_n ( - 2)^n \) (b) \( \sum_{n = 0}^{\infty} c_n ( - 4)^n \)

Step-by-Step Solution

Answer

(a) Yes, it converges. (b) Convergence cannot be assured.

Step 1: Understand convergence of the original series

The given series is \( \sum_{n = 0}^{\infty} c_n 4^n \). Since it's convergent, by the root test or ratio test for absolute convergence, it converges absolutely because any series of the form \( \sum_{n = 0}^{\infty} c_n x^n \) with \(|x| < R\) converges absolutely.

Step 2: Examining Series (a)

For \( \sum_{n=0}^{\infty} c_n (-2)^n \), the common ratio relative to the original series is \( \frac{-2}{4} = -\frac{1}{2} \). Since this series does not involve terms that include \( 4^n \) directly, we cannot automatically apply the same conclusion. However, by the comparison test, if \( |c_n(-2)^n| = |c_n|2^n \) converges since \( c_n 4^n \) converges, \( (-2)^n \) has a smaller base than \( 4^n \). Hence, this series converges.

Step 3: Examining Series (b)

For \( \sum_{n=0}^{\infty} c_n (-4)^n \), the common ratio relative to the original series is \( \frac{-4}{4} = -1 \). Notice that \((-4)^n = (-1)^n 4^n\), which means the terms are similar in magnitude to the original series but alternate in sign and are not assured to be absolutely convergent. By the alternating series test, we cannot directly conclude convergence unless \( c_n \) goes to zero suitably. Therefore, without more information about \( c_n \), the convergence of this series cannot be assured.

Key Concepts

Root Test

The root test is a helpful tool in determining the convergence of infinite series. It is especially useful when looking at series with terms raised to powers involving their indices. Suppose you have a series denoted as \( \sum_{n=0}^{\infty} a_n \). To apply the root test, you calculate the n-th root of the absolute value of the n-th term: \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \).

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), or if the limit is infinite, the series diverges.
• If \( L = 1 \), the test is inconclusive.

This test is particularly favorable for series involving exponential expressions. In the original exercise, if the series \( \sum_{n=0}^{\infty} c_n 4^n \) meets the condition \( |4| < R \), then it converges absolutely, making the root test effectively supportive in assessing its behavior.

Ratio Test

The ratio test is another technique for analyzing series convergence and is very similar to the root test. It uses the ratio of successive terms to deduce behavior. Consider a series \( \sum_{n=0}^{\infty} a_n \). You calculate: \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

In the provided problem solution, the ratio test could play a significant role in complementing the root test to ensure absolute convergence. It's particularly practical for series with factorials or other sequential elements where term-to-term comparison is straightforward, helping certify that \( \sum_{n=0}^{\infty} c_n 4^n \) converges absolutely.

Comparison Test

The comparison test is a simple but powerful method for determining the convergence of series, particularly when direct tests like the root test or ratio test fall short. The idea is to compare your series with another series that has known convergence properties.If you have series \( \sum_{n=0}^{\infty} a_n \) and \( \sum_{n=0}^{\infty} b_n \), for all \( n \):

• If \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
• If \( 0 \leq b_n \leq a_n \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.

In the initial exercise, for series (a), \( \sum_{n=0}^{\infty} c_n (-2)^n \) was shown to converge based on comparison with \( \sum_{n=0}^{\infty} c_n 4^n \), given \((-2)^n\) has a lower base, making it smaller in magnitude than \(4^n\), assuming \(c_n\) are non-negative terms. This method helps ensure the proper alignment of convergence properties.

Alternating Series Test

The alternating series test is pivotal when dealing with series that have terms alternating in sign, typically of the form \( (-1)^n a_n \). For the series \( \sum_{n=0}^{\infty} (-1)^n a_n \), the criteria for convergence are:

• The absolute value of the sequential term, \( a_n \), must be positive.
• \( a_n \) is a decreasing sequence, i.e., \( a_{n+1} \leq a_n \) for all \( n \) sufficiently large.
• \( a_n \) approaches zero as \( n \to \infty \).

In the problem statement, series (b) \( \sum_{n=0}^{\infty} c_n (-4)^n \) involves alternating terms. Here, we need more conditions beyond the absolute convergence to confirm the convergence due to its alternating property. If these conditions aren’t satisfied, we can't conclude its convergence simply based on the convergence of \( \sum_{n=0}^{\infty} c_n 4^n \) alone.