Question

Complete the proof of Theorem 4.5 .3 by showing that (a) \(R=0\) if \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right| /\left|a_{n}\right|=\infty ;\) (b) \(R=\infty\) if \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right| /\left|a_{n}\right|=0\).

Short answer

**Question:** Complete the proof of Theorem 4.5.3 by proving the following statements. (a) Prove that R=0 if the limit of the ratio of consecutive absolute terms approaches infinity as n approaches infinity. (b) Prove that R=∞ if the limit of the ratio of consecutive absolute terms approaches 0 as n approaches infinity. **Short Answer:** (a) We have shown that if the limit of the ratio of consecutive absolute terms (as n approaches infinity) approaches infinity, this means that the terms in the series grow without bound, and the convergence radius R approaches 0. (b) We have shown that if the limit of the ratio of consecutive absolute terms (as n approaches infinity) approaches 0, this means that the terms in the series shrink to 0, and the convergence radius R approaches infinity.

Step-by-step solution

Statement (a): Proof that R=0 if the limit of the ratio of consecutive absolute terms approaches infinity

To prove this statement, let's analyze the expression for the limit given in the exercise: \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right|/\left|a_{n}\right|=\infty.\) We will use the definition of convergence radius for a series: \(R = \lim\limits_{n\to\infty} \frac{1}{\lvert \frac{a_{n+1}}{a_n} \rvert} = \lim\limits_{n\to\infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert}.\) Now, since \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right|/\left|a_{n}\right|=\infty,\) this means that the denominator in our expression for R, i.e., \(\left|a_{n+1}\right|,\) grows without bound with respect to the numerator, \(\left|a_{n}\right|.\) This makes our expression for R to approach 0 as n approaches infinity: \(R = \lim\limits_{n\to\infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert} = 0.\) Thus, we have shown that \(R=0\) if \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right|/\left|a_{n}\right|=\infty .\)

Statement (b): Proof that R=∞ if the limit of the ratio of consecutive absolute terms approaches 0

To prove this statement, let's analyze the expression for the limit given in the exercise: \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right|/\left|a_{n}\right|=0.\) Recall the definition of convergence radius for a series: \(R = \lim\limits_{n\to\infty} \frac{1}{\lvert \frac{a_{n+1}}{a_n} \rvert} = \lim\limits_{n\to\infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert}.\) Now, since \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right|/\left|a_{n}\right|=0,\) this means that the numerator in our expression for R, i.e., \(\left|a_{n}\right|,\) grows without bound with respect to the denominator, \(\left|a_{n+1}\right|.\) This makes our expression for R to approach ∞ as n approaches infinity: \(R = \lim\limits_{n\to\infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert} = \infty.\) Thus, we have shown that \(R=\infty\) if \(\lim _{n \rightarrow \infty}\left|a_{n+1}\right|/\left|a_{n}\right|=0.\)

Key concepts

Limit of a Sequence

A fundamental concept in calculus and analysis, a sequence's limit involves understanding how a sequence behaves as the term number approaches infinity. Simply put, it helps us determine what value, if any, the terms of a sequence are getting closer to as the sequence progresses.
When we write \(\lim_{n \to \infty} a_n = L\), it signifies that as \(n\) becomes very large, the terms \(a_n\) approach the value \(L\).
Understanding whether this limit exists or not is crucial in examining the nature of series and functions at infinity.

• **Limit Involving Ratios:** In the context of the problem at hand, the limit involves a ratio of consecutive terms \(\lim_{n \to \infty} \left|a_{n+1}\right| / \left|a_{n}\right|\).
• When this ratio approaches infinity, the numerator dominates the denominator, suggesting that the sequence's magnitude is growing indefinitely.
• Conversely, if the ratio approaches zero, it indicates that the sequence's terms are becoming smaller and getting closer to zero as \(n\) increases.

Understanding these limits are vital for concluding about the radius of convergence in series.

Ratio Test for Series

The ratio test is a handy tool for determining whether a series converges absolutely or diverges. It can be succinctly explained with these steps:
1. Consider the series \(\sum a_n\) where \(n\) is a positive integer.
2. Observe the ratio \(\left| \frac{a_{n+1}}{a_n} \right|\) as \(n\) tends to infinity.
3. Define \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).

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

The ratio test gives a clear criterion for convergence by comparing the rate of growth of terms in the sequence.
This test's efficiency lies in simplifying complicated expressions by reducing them to evaluating limits, which is often more straightforward.
By analyzing the limit conditions given, we directly conclude about the radii of convergence in different scenarios.

Absolute Convergence

A series \(\sum a_n\) is said to converge absolutely if the series \(\sum |a_n|\) converges.
This concept extends the idea of convergence beyond just the plain series itself, accentuating the behavior of a series even when its individual terms are replaced by their absolute values.

• When a series converges absolutely, it guarantees convergence of the original series \(\sum a_n\), but not vice versa.
• The absolute convergence implies that rearranging terms doesn't affect the series' sum, a property not necessarily true for conditionally convergent series.
• The ratio test is particularly powerful in determining absolute convergence because it directly assesses the limit of ratios of absolute terms.

If the absolute value ratio of consecutive terms has a finite limit less than one, the series converges absolutely.
This concept is especially critical in the analysis of power series where the absolute convergence determines the boundaries of convergence.