Problem 76 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Show that the radius of convergence of the binomial series is 1 .

Step-by-Step Solution

Answer

The radius of convergence of the binomial series is 1.

Step 1 - Define the Binomial series

The general binomial series can be described as: \[ (1+x)^{\alpha} = 1 + \alpha x + \frac{\alpha (\alpha - 1)x^{2}}{2!} + \frac{\alpha (\alpha - 1)(\alpha - 2)x^{3}}{3!} + \dots\] where the coefficient of the nth term is \[a_{n} = \frac{\alpha(\alpha - 1) \dots (\alpha - n + 1)}{n!} x^{n}\]

Step 2 - Apply the Ratio Test

To find the radius of convergence, apply the Ratio Test. The Ratio Test requires solving for the absolute value of the limit of the ratio of the (n+1)th term to the nth term as n approaches infinity. So the ratio \(|a_{n+1}/a_{n}|\) becomes \( \left| \frac{ \frac{\alpha(\alpha - 1) \dots (\alpha - n + 1)}{n!} x^{n} }{ \frac{\alpha(\alpha - 1) \dots (\alpha - n)}{(n+1)!} x^{n+1} } \right|\). Simplify this to \( |x| \cdot \left| \frac{\alpha - n}{n+1} \right|\).

Step 3 - Take the limit

Find the limit of the absolute value of the ratio \( |x| \cdot \frac{\alpha - n}{n+1} \) as n approaches infinity. This limit gives \( |x| \cdot 0 \) which simplifies to 0. Therefore according to the ratio test, since the limit is less than 1, the series converges.

Step 4 - Find the radius of convergence

The radius of convergence is the reciprocal of the limit result, in this case 0. Since anything divided by 0 tends towards infinity, the interval of convergence is (-1, 1). The radius of convergence, being half the length of this interval, is r = 1.

Key Concepts

Binomial Series

The binomial series is a way of expanding expressions raised to any power, not just integers. It's written as:

• \[(1+x)^{\alpha} = 1 + \alpha x + \frac{\alpha (\alpha - 1)x^{2}}{2!} + \cdots\]

In this series, each term increases in power by 1 and involves a coefficient based on \(\alpha\), which can be any real number. This makes it flexible and widely used in mathematical calculations. The nth-term coefficient is given by:

• \[\frac{\alpha(\alpha - 1) \cdots (\alpha - n + 1)}{n!} x^{n}\]

Understanding the terms' coefficient is crucial, as they determine how rapidly each term grows or shrinks, impacting convergence.

Ratio Test

The Ratio Test is a method used to determine the convergence of a series. It's useful for series where each term is a function of the previous one, like the binomial series. The basic idea is to look at the ratio of consecutive terms to check their behavior as they move towards infinity.To apply the test:

• Calculate \(|a_{n+1}/a_{n}|\) for the series terms.
• Find the limit as n approaches infinity of this ratio.

For convergence, the limit must be less than 1.In our example, the ratio simplifies to:

• \(|x| \cdot \left| \frac{\alpha - n}{n+1} \right|\)

As \(n\) grows, \(\alpha - n\) becomes very negative, leading the limit to zero. Since 0 is less than 1, the series converges.

Convergence

Convergence in the context of series means that as we add more terms, the series approaches a finite value. It's important to understand whether a series converges, as it tells us about the stability and applicability of the series for different values.In the binomial series example:

• The limit \(|x| \cdot \frac{\alpha - n}{n+1}\) as \(n\) approaches infinity is 0.
• This is less than 1, indicating convergence within the interval of \(-1 < x < 1\).

Convergence ensures the series behaves well within this range, making calculations reliable.

Power Series

A power series is an infinite series of the form:

• \[\sum_{n=0}^{\infty} c_n x^n\]

Each term involves a power of \(x\), and these series are extremely useful in approximating functions. The convergence of a power series depends on \(x\) and the series coefficients.In the context of the binomial series:

• The series takes a power series form when expanded around 0.
• The convergence radius tells us the interval where the series gives accurate results.

By examining the power series, we can understand how well it approximates functions like \((1+x)^{\alpha}\) within a specific range.