Question

Suppose that \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=p .\) For which values of \(p\) must \(\sum_{n=1}^{\infty} 2^{n} a_{n}\) converge?

Short answer

The series converges for \(p < \frac{1}{2}\).

Step-by-step solution

Understanding the problem

We are tasked with determining for which values of \(p\) the series \(\sum_{n=1}^{\infty} 2^{n} a_{n}\) converges, given that \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=p\). This is essentially asking about the convergence behavior of a series with terms \(2^{n} a_{n}\) based on the ratio limit of the sequence \(a_n\).

Applying the Ratio Test

The ratio test gives us a tool to determine the convergence of a series \(\sum_{n=1}^{\infty} b_n\), where the absolute value \(\lim_{n \to \infty} \left|\frac{b_{n+1}}{b_n}\right| = L\). If \(L < 1\), the series converges. We apply this to \(b_n = 2^n a_n\).

Finding the Expression for the Ratio

Consider the term \(b_n = 2^n a_n\). Then, the ratio is \(\frac{b_{n+1}}{b_n} = \frac{2^{n+1} a_{n+1}}{2^n a_n} = 2 \cdot \frac{a_{n+1}}{a_n}\). Thus, \(\left|\frac{b_{n+1}}{b_n}\right| = 2 \cdot \left|\frac{a_{n+1}}{a_n}\right|\).

Substituting the Limit Hypothesis

Since \(\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = p\), we have \(\lim_{n \to \infty} 2 \cdot \left|\frac{a_{n+1}}{a_n}\right| = 2p\).

Determining Convergence Condition

For the series \(\sum_{n=1}^{\infty} 2^n a_n\) to converge, the ratio test requires \(2p < 1\). Solving \(2p < 1\) gives us the condition \(p < \frac{1}{2}\).

Concluding the Values of p

Thus, the series \(\sum_{n=1}^{\infty} 2^n a_n\) converges for values of \(p\) such that \(p < \frac{1}{2}\).

Key concepts

Ratio Test

The Ratio Test is a powerful technique used to determine the convergence of an infinite series. This test assesses the behavior of the ratio of successive terms in the series as the sequence progresses. If we have a series with general term \(b_n\), the Ratio Test involves evaluating the limit \(L = \lim_{n \to \infty} \left|\frac{b_{n+1}}{b_n}\right|\). The outcome of the test provides a decisive conclusion regarding the convergence of the series:

• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive, and convergence cannot be determined using this method.

For the given problem, the series \(\sum_{n=1}^{\infty} 2^n a_n\) uses this test. The ratio of terms \(\frac{b_{n+1}}{b_n} = \frac{2^{n+1} a_{n+1}}{2^n a_n} = 2 \cdot \frac{a_{n+1}}{a_n}\) simplifies using the initial ratio limit \( \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| = p \). Therefore, evaluating \( \lim_{n \to \infty} \left|\frac{b_{n+1}}{b_n}\right| = 2p\), the series converges if \(2p < 1\). This calculates to the condition \(p < \frac{1}{2}\), indicating the series converges for such values of \(p\).

Limit of Sequence

The limit of a sequence is a fundamental concept in calculus, describing the behavior of sequences as they approach infinity. For any sequence \(a_n\) and its limit \(\lim_{n \to \infty} a_n = L\), if the sequence converges, it gets arbitrarily close to the value \(L\) as \(n\) becomes very large.

In this problem, we consider a sequence \(a_n\) where \( \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = p \). This is a specific type of limit referring to the growth rate of the sequence terms relative to each other. The value \(p\) indicates whether the terms \(a_{n+1}\) increase or decrease compared to \(a_n\) as \(n\) grows. When examining a series for convergence, these ratio limits are crucial in determining if the series will have a finite sum as more terms are included. As explored, the limit influences the outcome of the Ratio Test application, vital for understanding the convergence conditions of the series \(\sum_{n=1}^{\infty} 2^n a_n\).

Infinite Series Convergence

Infinite series convergence refers to whether the sum of an infinite sequence of terms approaches a finite value. This idea is central in many areas of mathematics, including calculus and analysis. Convergence involves ensuring that as more terms are added, they eventually contribute less and less to the total sum.

To assess convergence, several tests can be employed, the Ratio Test being one of the most common for series. Particularly, when dealing with power series or exponential terms, the Ratio Test helps by examining the ratio of consecutive terms. Here, the infinite series \(\sum_{n=1}^{\infty} 2^n a_n\) was analyzed using this test. Given \( \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = p \), applying the condition \(2p < 1\) determines that the series converges for \(p < \frac{1}{2}\).

Understanding these convergence criteria is essential for students, as it forms the basis for more complex topics related to series, functions, and their applications in science and engineering. Ensuring the rigorous application of tests such as the Ratio Test ensures correct judgments about the behavior of series.