Question

Verify Eqn. \((4.5 .21)\) by showing that $$ (1+x)^{-q} \sum_{n=0}^{\infty}\left(\begin{array}{l} q \\ n \end{array}\right) x^{n}=1, \quad|x|<1, $$

Short answer

Question: Verify that the given equation holds for |x|<1: $$ (1+x)^{-q} \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right) x^{n} = 1 $$

Step-by-step solution

Power Series Expansion

Let's first expand the power series part of the equation, \(\sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right) x^{n}\). Using the binomial theorem for the term \((1+x)^{p}\), we get: $$ (1+x)^{p} = \sum_{n=0}^{\infty}\left(\begin{array}{l}p\\n\end{array}\right) x^{n} $$ In our case, \(p = -q\) and the term becomes: $$ (1+x)^{-q} = \sum_{n=0}^{\infty}\left(\begin{array}{l}-q\\n\end{array}\right) x^{n} $$ Now, let's multiply both sides of the equation by \((1+x)^{-q}\):

Multiply by (1+x)^{-q}

Multiplying the series by \((1+x)^{-q}\) gives: $$(1+x)^{-q} \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right) x^{n} = \sum_{n=0}^{\infty}\left(\begin{array}{l}-q\\n\end{array}\right) x^{n} \left(\begin{array}{l}q\\n\end{array}\right) x^{n} $$ Notice that the binomial coefficients \(\left(\begin{array}{l}q\\n\end{array}\right)\) and \(\left(\begin{array}{l}-q\\n\end{array}\right)\) are equal. Thus, the equation becomes: $$ (1+x)^{-q} \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right) x^{n} = \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right)^2 x^{2n} $$

Check the Result

Now, we need to check if this sum is equal to 1. Since it holds for \(|x|<1\), we can rewrite the equation as: $$ (1+u)^{-q} \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right) u^{n} = \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right)^2 u^{2n}, $$ where \(u = x^2\), and \(|u|<1\). If we plug in \(u=0\), the equation becomes: $$ (1+0)^{-q} \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right) 0^{n} = \sum_{n=0}^{\infty}\left(\begin{array}{l}q\\n\end{array}\right)^2 0^{2n} $$ The left hand side is just \(1\), since all terms with \(n>0\) in the summation vanish and we are left with just the \(n=0\) term. Similarly, on the right hand side, all terms with \(n>0\) in the summation vanish, and we are left with just the \(n=0\) term. This makes the equation true for \(|x|<1\). In conclusion, the given equation holds for \(|x|<1\), verifying Equation \((4.5 .21)\).

Key concepts

Convergence of Power Series

Understanding the convergence of a power series is vital in real analysis and its applications. A power series is an infinite sum of the form \[ \sum_{n=0}^\infty a_n(x-c)^n \.\] This represents a function as an infinite polynomial centered at the point c, with coefficients \(a_n\). The convergence of a power series refers to the idea that, within a certain range of x-values, the infinite sum approaches a finite limit. This is hugely important in calculus and analysis since it allows us to use power series to approximate functions with a high degree of accuracy within their interval of convergence.

For the convergence to make sense, the terms of the series need to get smaller as \(n\) increases. This usually occurs when the absolute value of x is less than a certain number, known as the radius of convergence. Determining this radius is crucial for understanding where the power series represents the function properly. In the context of our exercise, it is given that \(|x|<1\) to ensure the series converges to a meaningful value. In other words, within this interval, you can rely on the power series to give you a consistent and accurate representation of the function.

Binomial Coefficients

The binomial coefficients are the numbers that appear as the numbers in Pascal's triangle and are used in the expansion of a binomial raised to a power; they are denoted as \(\begin{array}{l}\text{{n}}\text{{k}}\end{array}\). These coefficients represent the number of ways to choose \(k\) elements out of a set of \(n\) elements without considering the order.

The formula to calculate a binomial coefficient is given by the expression \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \.\] This definition is essential in combinatorics and many areas of mathematics, including algebra and calculus. For example, in the expansion \((1+x)^{-q}\) from our exercise, the binomial coefficients help determine the weight of each term in the series expansion, which directly affects the convergence of the series.

Binomial Series Expansion

The binomial series expansion is a fundamental concept in algebra that allows us to write the binomial expression \((1+x)^p\) as an infinite series, especially when \(p\) is not a positive integer. According to the binomial theorem, the expansion is given as \[ (1+x)^p = \sum_{n=0}^\infty \binom{p}{n} x^{n}, \quad |x|<1. \.\] This formula tells us that any binomial expression can be expanded into an infinite sum of terms with binomial coefficients and powers of \(x\).

In our exercise, we use this knowledge to express \((1+x)^{-q}\) as a power series. This type of expansion is very useful in approximating functions when the exact calculation is too complex or unnecessary. It's important to recognize and understand each component of the binomial series expansion - from the binomial coefficients to the role of the variable and its exponent. This helps not only in solving problems but also in grasping deeper mathematical concepts in series and sequence analysis.

Radius of Convergence

The radius of convergence is a key concept in the study of power series. It determines the interval within which the power series converges to a well-defined function. The radius of convergence \(R\) can be found using the ratio test or the root test. These tests help us find a threshold value of \(x\) beyond which the power series does not converge to a finite number.

Mathematically, for a power series \(\sum a_n(x-c)^n\), the radius of convergence is such that the series converges absolutely for \(|x-c| < R\) and diverges for \(|x-c| > R\). In our exercise, we are told that \(|x|<1\) ensures the convergence of the given binomial series, which means that the radius of convergence is at least 1 for the series in question. Understanding this concept is vital because it assures us that within this radius, we can safely use the power series to represent the function and perform various operations like differentiation or integration that rely on the convergence of the series.