Problem 6
Question
Let \(\sum_{n=0}^{\infty} a_{n} z^{n}\) be a power series with radius of convergence \(r\). Show: (a) If \(\lim _{n \rightarrow \infty}\left|a_{n}\right| /\left|a_{n+1}\right|\) exists, then \(r=R\). (b) If \(\bar{\rho} \lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} \in[0, \infty]\) exists, then \(r=1 / \bar{\rho} .\) Here we formally use the conventions \(1 / 0=\infty\) and \(1 / \infty=0 .(r=0\) for \(\bar{\rho}=\infty\), and \(r=\infty\) for \(\bar{\rho}=0 .)\) (c) If we set $$ \rho:=\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}:=\lim _{n \rightarrow \infty}\left(\sup \left\\{\sqrt[n]{\left|a_{n}\right|}, \sqrt[n+1]{\left|a_{n+1}\right|}, \sqrt[n+1]{\left|a_{n+1}\right|}, \ldots\right\\}\right) $$ then following the same conventions as in (b) there holds: \(r=1 / \rho\) (A.-L. CAUCHY, 1821;J. HADAMARD, 1892 )
Step-by-Step Solution
Verified Answer
(a) \( r = R \). (b) \( r = 1/\bar{\rho} \). (c) \( r = 1/\rho \).
1Step 1: Identify the Limit of Ratio Test (Part a)
To solve (a), recognize that if the limit \( \lim _{n \rightarrow \infty}\left|a_{n}\right| /\left|a_{n+1}\right| \) exists, this aligns with the Ratio Test for convergence of power series. According to this test, the radius of convergence \( r \) satisfies \( r = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| = R \), where \( R \) is the limit of these ratios. This confirms \( r = R \).
2Step 2: Analyze the Root Test for Convergence (Part b)
For part (b), if \( \bar{\rho} = \lim _{n \rightarrow \infty} \sqrt[n]{\left| a_{n} \right|} \) exists, the Root Test implies the radius of convergence \( r = \frac{1}{\bar{\rho}} \). Here, the conventions \( 1/0 = \infty \) and \( 1/\infty = 0 \) indicate that if \( \bar{\rho} = \infty \), then \( r = 0 \), and if \( \bar{\rho} = 0 \), then \( r = \infty \).
3Step 3: Application of the Lim Sup Definition (Part c)
In part (c), the definition \( \rho = \lim _{n \rightarrow \infty} \left( \sup \left\{ \sqrt[n]{\left| a_{n} \right|}, \sqrt[n+1]{\left| a_{n+1} \right|}, \ldots \right\} \right) \) refines part (b). \( \lim \sup \) involves the greatest limit point of the sequence, reinforcing that the radius of convergence \( r = \frac{1}{\rho} \). Following the same conventions, if \( \rho = \infty \), then \( r = 0 \), and if \( \rho = 0 \), then \( r = \infty \).
Key Concepts
Ratio TestRoot TestPower SeriesLimit SuperiorConvergence Criteria
Ratio Test
The Ratio Test is a valuable tool for determining the radius of convergence of a power series. When given a series expressed as \(\sum_{n=0}^{\infty} a_{n} z^{n}\), the Ratio Test evaluates the limit \(\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right|\). This limit helps us understand how the coefficients \(a_n\) behave as \(n\) increases.
If this limit exists and is finite, the power series converges when \(|z| < r\) where \(r\) is the radius of convergence. In essence, if our limit \(\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right| = R\), then \(r = R\). This means that for \(|z|
If this limit exists and is finite, the power series converges when \(|z| < r\) where \(r\) is the radius of convergence. In essence, if our limit \(\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right| = R\), then \(r = R\). This means that for \(|z|
Root Test
The Root Test is another powerful method for determining the convergence of a power series. This test focuses on the \(n\)-th root of the absolute value of the terms. For a power series \(\sum_{n=0}^{\infty} a_{n} z^{n}\), we use the sequence \(\sqrt[n]{\left| a_{n} \right|}\).
The limit \(\lim _{n \rightarrow \infty} \sqrt[n]{\left| a_{n} \right|} = \bar{\rho}\) gives us the information needed to find the radius of convergence, \(r\). Here, the formula \(r = \frac{1}{\bar{\rho}}\) is used. Following this logic, if \(\bar{\rho} = 0\), intuition tells us that \(r\) becomes infinity, making the power series absolutely convergent for all \(z\). Conversely, if \(\bar{\rho} = \infty\), \(r = 0\), indicating convergence only at \(z = 0\).
The limit \(\lim _{n \rightarrow \infty} \sqrt[n]{\left| a_{n} \right|} = \bar{\rho}\) gives us the information needed to find the radius of convergence, \(r\). Here, the formula \(r = \frac{1}{\bar{\rho}}\) is used. Following this logic, if \(\bar{\rho} = 0\), intuition tells us that \(r\) becomes infinity, making the power series absolutely convergent for all \(z\). Conversely, if \(\bar{\rho} = \infty\), \(r = 0\), indicating convergence only at \(z = 0\).
- \(\bar{\rho} = 0\) implies \(r = \infty\)
- \(\bar{\rho} = \infty\) implies \(r = 0\)
- \(0 < \bar{\rho} < \infty\) implies \(r = \frac{1}{\bar{\rho}}\)
Power Series
A power series is an infinite series of the form \(\sum_{n=0}^{\infty} a_{n} z^{n}\), where \(a_n\) are coefficients and \(z\) is a complex variable. The concept of power series is crucial because it generalizes the idea of a polynomial to infinite terms.
The radius of convergence \(r\) defines the interval or disk in which the series converges absolutely. Within this region \(|z| < r\), the series behaves nicely and sums up to a finite value. Similarly, the boundary and beyond this interval, convergence behavior needs to be analyzed separately, as the series may diverge or conditionally converge.
Power series are used widely in mathematics, particularly in topics like analytic functions and complex analysis. Understanding the behavior of such series is pivotal in solving differential equations and performing integrations, thanks to their ability to represent functions as sums of powers of \(z\).
The radius of convergence \(r\) defines the interval or disk in which the series converges absolutely. Within this region \(|z| < r\), the series behaves nicely and sums up to a finite value. Similarly, the boundary and beyond this interval, convergence behavior needs to be analyzed separately, as the series may diverge or conditionally converge.
Power series are used widely in mathematics, particularly in topics like analytic functions and complex analysis. Understanding the behavior of such series is pivotal in solving differential equations and performing integrations, thanks to their ability to represent functions as sums of powers of \(z\).
Limit Superior
Limit superior, often abbreviated as \(\lim \sup\), is a concept used to analyze the limiting behavior of a sequence. It refers to the smallest number that serves as an upper bound of the sequence's limit points.
Practically, when applied to power series, \(\lim \sup\) can be used to refine convergence tests such as the Root Test. For a sequence \(\{a_n\}\), the notation \(\lim _{n \rightarrow \infty} \sup \) indicates taking the greatest accumulation point of the sequence's \( n \)-th root terms.
This concept becomes particularly crucial when dealing with series that have terms with non-standard growth rates or oscillations.
Practically, when applied to power series, \(\lim \sup\) can be used to refine convergence tests such as the Root Test. For a sequence \(\{a_n\}\), the notation \(\lim _{n \rightarrow \infty} \sup \) indicates taking the greatest accumulation point of the sequence's \( n \)-th root terms.
- Helps in determining\(\overline{\rho}\), which is vital for finding the radius of convergence.
- Assists in managing variations in sequence terms that may not converge traditionally.
This concept becomes particularly crucial when dealing with series that have terms with non-standard growth rates or oscillations.
Convergence Criteria
Convergence criteria for power series rely on understanding whether the series converges or diverges based on its given terms. Here, tests like the Ratio Test and Root Test play pivotal roles.
A power series \(\sum_{n=0}^{\infty} a_{n} z^{n}\) is said to converge within its radius of convergence, \(r\). This means for any \(|z| < r\), the series converges absolutely. However, the behavior at \(|z| = r\) requires separate analysis, while for \(|z| > r\), divergence is certain.
Key convergence criteria include:
Both criteria aim to provide clear guidelines on determining the set of values \(z\) for which the power series will converge neatly, making them indispensable tools for analyzing series.
A power series \(\sum_{n=0}^{\infty} a_{n} z^{n}\) is said to converge within its radius of convergence, \(r\). This means for any \(|z| < r\), the series converges absolutely. However, the behavior at \(|z| = r\) requires separate analysis, while for \(|z| > r\), divergence is certain.
Key convergence criteria include:
- The Ratio Test where \(\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right|\)
- The Root Test with the limit \(\lim _{n \rightarrow \infty} \sqrt[n]{\left| a_{n} \right|}\)
Both criteria aim to provide clear guidelines on determining the set of values \(z\) for which the power series will converge neatly, making them indispensable tools for analyzing series.
Other exercises in this chapter
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