Problem 30 - Calculus

Question

(a) Write the general term of the binomial series for \((1+x)^{p}\) about \(x=0\). (b) Find the radius of convergence of this series.

Step-by-Step Solution

Answer

(a) General term: \(a_n = \frac{p(p-1)...(p-n+1)}{n!} x^n\); (b) Radius of convergence: 1.

Step 1: Understanding the Binomial Series

The binomial series for \((1+x)^p\) can be written as a series of terms derived from the expansion \((1+x)^p = \sum_{n=0}^{\infty} \binom{p}{n} x^n\), where \(\binom{p}{n}\) represents the generalized binomial coefficient.

Step 2: Write the General Term

The coefficient \(\binom{p}{n}\) in the series is defined as \(\binom{p}{n} = \frac{p(p-1)(p-2)...(p-n+1)}{n!}\) for \(n \geq 0\). Therefore, the general term of the binomial series is \(a_n = \binom{p}{n} x^n = \frac{p(p-1)(p-2)...(p-n+1)}{n!} x^n\).

Step 3: Determine the Radius of Convergence Using Ratio Test

To find the radius of convergence, apply the ratio test. Compute \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\binom{p}{n+1} x^{n+1}}{\binom{p}{n} x^n} \right|\). This simplifies to \(\lim_{n \to \infty} \left| \frac{(p-n)}{n+1} x \right|\).

Step 4: Solve the Limit for Convergence Radius

The series converges when this limit is less than 1: \(\lim_{n \to \infty} \left| \frac{(p-n)}{n+1} x \right| < 1\). As \(n\) becomes very large, \(\lim_{n \to \infty} \left| \frac{(p-n)}{n+1} x \right| \approx \left| x \right| < 1\). Therefore, the radius of convergence is \(R = 1\).

Key Concepts

Binomial Theorem

In mathematics, the Binomial Theorem is a powerful tool used to expand expressions of the form \((1+x)^p\) into a series. This series is characterized by terms that include powers of \(x\) multiplied by binomial coefficients. The general form for this expansion is:

• \[(1+x)^p = \sum_{n=0}^{\infty} \binom{p}{n} x^n\]

Here, \(\binom{p}{n}\) is referred to as the generalized binomial coefficient. It is an extension of the binomial coefficients you may know from Pascal’s Triangle. Specifically, the generalized coefficient is calculated by:

• \[\binom{p}{n} = \frac{p(p-1)(p-2)\ldots(p-n+1)}{n!}\]

This formula allows for series expansions even when \(p\) is not a non-negative integer. Each term in the series thus consists of these coefficients multiplied by the corresponding power of \(x\). This expansion is fundamental when dealing with series in calculus and helps in approximations of functions.

Radius of Convergence

Finding the radius of convergence is essential to understanding the range of values for which a series converges. For the binomial series, or any power series, the radius of convergence \(R\) determines where the series converges when we input different values of \(x\). To determine the radius of convergence for \((1+x)^p\), we often use the Ratio Test. By examining the ratio of successive terms, we find:

• \[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(p-n)}{n+1} x \right|\]

The series converges absolutely when this limit is less than 1, implying \(\left| x \right| < 1\). Thus, the radius of convergence for the binomial series is \(R = 1\). This signifies that the series converges for values of \(x\) within the interval \((-1, 1)\). Any value of \(x\) outside this range may result in divergence.

Ratio Test

The Ratio Test is a method used to determine the convergence of an infinite series. It is particularly useful for power series like the binomial series. The test involves taking the limit of the absolute value of the ratio of consecutive terms:

• \[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]

For convergence, this limit must be less than 1. Applying the Ratio Test to the binomial expansion term \(a_n = \binom{p}{n} x^n\), results in:

• \[\lim_{n \to \infty} \left| \frac{(p-n)}{n+1} x \right| < 1\]

As we simplify, the critical condition for convergence emerges as \(\left| x \right| < 1\), which directly informs us about the radius of convergence. Although powerful, the Ratio Test sometimes yields a result of 1, which means it is inconclusive, and other tests may need to be applied. However, for many series, including the one discussed here, it efficiently determines where the series converges.