Question

Verify the first four terms of each infinite binomial series. $$1 / \sqrt[6]{1-a}=1+a / 6+7 a^{2} / 72+91 a^{3} / 1296 \ldots$$

Short answer

The first four terms of the binomial series for \(1 / \sqrt[6]{1-a}\) are confirmed to be \(1\), \(a / 6\), \(7 a^{2} / 72\), and \(91 a^{3} / 1296\).

Step-by-step solution

Understand the Binomial Series

For a binomial expression of the form \( (1 - x)^n \) where \( n \) is a real number, and \( |x| < 1 \), the infinite binomial series can be expanded as: \[ (1 - x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \] In this exercise, we are working with the expression \( 1 / \sqrt[6]{1-a} = (1 - a)^{-1/6} \), so we will expand it using the binomial series expansion with \( n = -1/6 \) and \( x = a \).

First Term of the Series

The first term of the binomial series is always \( 1 \) since \((1 - a)^0 = 1\), regardless of the value of \( a \) or \( n \).

Second Term of the Series

The second term is obtained using the linear term of the binomial expansion: \[ nx = \frac{-1}{6}a \] Hence, the second term is \( \frac{a}{6} \).

Third Term of the Series

The third term involves the quadratic term of the expansion: \[ \frac{n(n-1)}{2!}x^2 = \frac{-1/6(-1/6 - 1)}{2}a^2 = \frac{-1/6(-7/6)}{2}a^2 = \frac{7a^2}{72} \] So, the third term is \( \frac{7a^2}{72} \).

Fourth Term of the Series

The fourth term is the cubic term of the expansion: \[ \frac{n(n-1)(n-2)}{3!}x^3 = \frac{-1/6(-1/6 - 1)(-1/6 - 2)}{6}a^3 = \frac{-1/6(-7/6)(-13/6)}{6}a^3 = \frac{91a^3}{1296} \] Hence, the fourth term is \( \frac{91a^3}{1296} \).

Key concepts

Infinite Series

Understanding the concept of an infinite series is crucial in calculus and advanced mathematics. An infinite series is essentially a sum of an infinite number of terms that follow a certain pattern or rule. These terms may represent numbers, functions, or even more complex mathematical objects.

When dealing with infinite series, it's essential to know about convergence and divergence. A series converges if the sum of its terms approaches a specific value as more terms are added. Conversely, if the sum does not approach any limit and instead grows without bound, the series diverges.

In the context of the binomial series expansion, the series we're looking at is:

\[1 / \sqrt[6]{1-a} = 1 + a / 6 + 7 a^{2} / 72 + 91 a^{3} / 1296 \ldots\]
This is an example of a binomial series that can be expanded indefinitely and is expected to converge for certain values of \( a \), specifically where \( |a|<1 \). Such constraints ensure that the infinite series will provide meaningful and finite results.

Calculus

Calculus, the mathematical study of continuous change, is fundamentally intertwined with the concept of infinite series. It is in calculus that students typically encounter series and learn to work with them through methods of differential and integral calculus.

For instance, the sum of an infinite series is often associated with an integral, particularly when dealing with functions represented by power series. Similarly, the idea of deriving a function to find its rate of change also applies to series, where you can consider the derivative of the sum as the sum of the derivatives.

When verifying the terms of a binomial series, like in the given exercise, calculus is applied to both understand the patterns of the terms and to investigate their behaviors. For example, the binomial series expansion involves factorials and powers that come directly from combinatorics and polynomial expansions, which are key topics within calculus:

Binomial Expansion in Calculus:

\[(1 - x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots\]
This formula is a cornerstone in calculus, allowing the calculation and approximation of powers of binomials that may seem daunting at first glance.

Mathematical Series

The term mathematical series encompasses a wide array of sequences and sums. It's a broad category including geometric series, arithmetic series, harmonic series, and, of particular interest here, binomial series.

Each type of mathematical series has a unique set of characteristics and rules. For example, a geometric series is based on a constant ratio between terms, while an arithmetic series has a constant difference. Binomial series, however, expand a binomial expression into an infinite number of terms following the binomial theorem.

The exercise in question involves a binomial series which can be thought of as a power series with a particular form. It represents an expansion of a binomial raised to any type of power, including fractional and negative powers:

Application of Binomial Series:

In practice, a series such as \(1 / \sqrt[6]{1-a}\) is useful in fields like physics and engineering, where series can model natural phenomena and solutions to differential equations. When expanded, they offer a practical way to approximate functions and solutions to an arbitrary accuracy, by considering a finite number of terms.