Question

What tests are used to determine the radius of convergence of a power series?

Short answer

Question: List and briefly explain three tests used to determine the radius of convergence of a power series. Answer: The three main tests to determine the radius of convergence of a power series are: 1. Ratio Test: In this test, the radius of convergence is found by calculating the limit of the ratio of consecutive terms as \(n\) approaches infinity. \[R = \lim_{n \rightarrow \infty} \frac{|a_n|}{|a_{n+1}|}\] 2. Root Test: In the Root Test, the radius of convergence is obtained by taking the reciprocal of the limit of the \(n\)th root of the absolute value of the sequence's terms as \(n\) approaches infinity. \[R = \lim_{n \rightarrow \infty} \left(\frac{1}{(\sqrt[n]{|a_n|})}\right)\] 3. Cauchy-Hadamard Theorem: According to this theorem, the radius of convergence is given by the reciprocal of the limit superior of the sequence of the \(n\)th root of the absolute value of the terms, as \(n\) approaches infinity. \[R = \frac{1}{\limsup_{n \rightarrow \infty} (\sqrt[n]{|a_n|})}\] All three tests help determine the range within which the given power series converges.

Step-by-step solution

1. Introduction to Power Series

Power series are infinite series of the form: \[\sum_{n=0}^{\infty}a_n(x-c)^n\] where \(a_n\), \(x\), and \(c\) are real or complex numbers. The radius of convergence (\(R\)) is a non-negative number that defines the interval or disc within which the power series converges. To determine the radius of convergence for a given power series, we can use different tests.

2. Ratio Test

One of the tests used to determine the radius of convergence is the Ratio Test. According to this test, for a power series \(\sum_{n=0}^{\infty}a_n(x-c)^n\), the radius of convergence is defined by: \[R = \lim_{n \rightarrow \infty} \frac{|a_n|}{|a_{n+1}|}\] Calculate the limit for the series, and if it exists, the series converges for all values of \(x\) within the range \((c-R, c+R)\).

3. Root Test

Another test used to find the radius of convergence is the Root Test. In this test, for a power series \(\sum_{n=0}^{\infty}a_n(x-c)^n\), the radius of convergence is defined by: \[R = \lim_{n \rightarrow \infty} \left(\frac{1}{(\sqrt[n]{|a_n|})}\right)\] Calculate the limit for the series, and if it exists, the series converges for all values of \(x\) within the range \((c-R, c+R)\).

4. Cauchy-Hadamard Theorem

Cauchy-Hadamard theorem is another powerful tool to determine the radius of convergence. It states that for a power series \(\sum_{n=0}^{\infty} a_n (x-c)^n\), the radius of convergence is given by: \[R = \frac{1}{\limsup_{n \rightarrow \infty} (\sqrt[n]{|a_n|})}\] where \(\limsup_{n \rightarrow \infty}\) is the limit superior as \(n\) tends to infinity. Calculate the limit superior of the sequence \(\sqrt[n]{|a_n|}\), and then use the formula above to determine the radius of convergence. In summary, there are three main tests to find the radius of convergence of a power series: Ratio Test, Root Test, and Cauchy-Hadamard Theorem. These tests help to determine the range or disc within which the given power series converges.

Key concepts

Power Series

To start off, let's delve into the world of power series. A power series is like a polynomial extended into the infinite, where the powers increase without end. Specifically, it can be expressed in the form
\[\sum_{n=0}^{\infty}a_n(x-c)^n\]
Here, each term involves the coefficient \(a_n\), the variable \(x\), and a constant \(c\), which is the series' center. But unlike a simple polynomial, this series stretches on infinitely. One intriguing aspect of power series is their convergence - that means, under which conditions the endless sum actually reaches a finite value. This depends heavily on the value of \(x\) and is precisely defined within an interval or disk centered around \(c\), known as the radius of convergence (\(R\)). Understanding this radius is crucial for many mathematical and practical applications, as it tells us where the power series can be reliably used to represent functions.

Ratio Test

The Ratio Test is a sleek, yet powerful technique for analyzing convergence. When you're faced with a power series and yearning to discover its radius of convergence, the Ratio Test should be one of your first ports of call. Here's how it goes: you calculate the limit of the absolute value ratio of consecutive coefficients as \(n\) approaches infinity:
\[R = \lim_{n \rightarrow \infty} \frac{|a_n|}{|a_{n+1}|}\]
When you get a finite number, that's your radius of convergence, \(R\). Basically, the Ratio Test provides us with a straightforward method for finding the sweet spot - the range of \(x\) values - within which the series will gloriously converge.

Root Test

Moving onto the Root Test, imagine you're a mathematical detective, sussing out the range where a power series behaves. The Root Test is another crucial tool in your investigative kit. It involves taking the nth root of the absolute value of the coefficients, and as before, sending \(n\) off towards infinity:
\[R = \lim_{n \rightarrow \infty} \left(\frac{1}{(\sqrt[n]{|a_n|})}\right)\]
What emerges from this limit—if it exists—is the radius of convergence, telling you yet again about the interval where the series plays nice. It's especially handy in cases where the Ratio Test might be inconclusive or difficult to apply.

Cauchy-Hadamard Theorem

Rounding things out, let's look at the prestigious Cauchy-Hadamard Theorem, a true classic when it comes to dissecting power series. This theorem moves beyond the ratios and roots, introducing the concept of the limit superior (or lim sup for short). Here’s the theorem’s elegant formula:
\[R = \frac{1}{\limsup_{n \rightarrow \infty} (\sqrt[n]{|a_n|})}\]
In simple words, you determine the lim sup of the sequence involving the nth root of the absolute coefficient values. Then, take its reciprocal to get the radius of convergence, \(R\). The Cauchy-Hadamard Theorem shines with its generality, giving you a robust and applicable formula in scenarios where other tests may be complex or not yield clear results.