Problem 9 - Thomas Calculus

Question

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1+\frac{1}{x}\right)^{1 / 2} \end{equation}

Step-by-Step Solution

Answer

The first four terms are: \(1 + \frac{1}{2x} - \frac{1}{8x^2} + \frac{1}{16x^3}\).

Step 1: Understanding the Binomial Series

The binomial series for \((1 + x)^n\) is an infinite series given by \((1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k\), where \(\binom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!}\). In this particular problem, we substitute \(n = \frac{1}{2}\) and \(x = \frac{1}{x}\). We will find the first four terms of this series.

Step 2: Finding the First Term

For \(k = 0\), the term of the series is \(\binom{\frac{1}{2}}{0} \left(\frac{1}{x}\right)^0 = 1\). This is because any number to the power of zero is 1 and \(\binom{n}{0} = 1\).

Step 3: Calculating the Second Term

For \(k = 1\), we compute the term: \(\binom{\frac{1}{2}}{1} \left(\frac{1}{x}\right)^1 = \frac{1}{2} \cdot \frac{1}{x} = \frac{1}{2x}\).

Step 4: Determining the Third Term

For \(k = 2\), the term is computed as follows: \(\binom{\frac{1}{2}}{2} \left(\frac{1}{x}\right)^2 = \frac{\frac{1}{2} \cdot (-\frac{1}{2})}{2} \cdot \frac{1}{x^2} = -\frac{1}{8x^2}\).

Step 5: Calculating the Fourth Term

For \(k = 3\), the term is: \(\binom{\frac{1}{2}}{3} \left(\frac{1}{x}\right)^3 = \frac{\frac{1}{2} \cdot (-\frac{1}{2}) \cdot (-\frac{3}{2})}{6} \cdot \frac{1}{x^3} = \frac{1}{16x^3}\).

Step 6: Conclusion

Combining all these terms, the first four terms of the binomial series for \(\left(1 + \frac{1}{x}\right)^{1/2}\) are: \(1 + \frac{1}{2x} - \frac{1}{8x^2} + \frac{1}{16x^3}\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool that allows us to expand expressions of the form \((1 + x)^n\) into a series. This is particularly useful when dealing with powers that are not whole numbers. The binomial theorem states that:

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

The symbol \(\binom{n}{k}\) represents "n choose k," a binomial coefficient calculated as:

• \(\binom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!}\)

This coefficient tells us how many ways we can choose \(k\) items from \(n\) items without regard to order. When we expand a binomial using this theorem, we calculate each term separately by substituting the desired values of \(n\) and \(x\) into each term of the infinite series. It is also important for calculating binomial expansions with fractional exponents, like the problem presented in this exercise.

Series Expansion

A series expansion involves expressing a function as a sum of terms. In mathematics, series expansions are vital for simplifying complex functions so they can be analyzed more easily. The series expansion of the function \(\left(1+\frac{1}{x}\right)^{1/2}\) is found by applying the binomial series formula, where each term is obtained by substituting values of \(k\) into the series formula.
Analyzing each term:

• The first term corresponds to \(k=0\), which is \(1\).
• The second term \(\frac{1}{2x}\) comes from \(k=1\).
• The third and fourth terms are \(-\frac{1}{8x^2}\) and \(\frac{1}{16x^3}\) obtained by setting \(k=2\) and \(k=3\), respectively.

To compute each term, you calculate the binomial coefficient for each \(k\) and multiply it by \(\left(\frac{1}{x}\right)^k\). This approach breaks down complex power functions into more manageable expressions for computation.

Mathematical Induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. It is not directly used in calculating binomial series, but understanding it helps in validating many aspects of sequences and series. Induction follows a two-step process:

• **Base case**: Verify the statement for the initial value, usually \(n=0\) or \(n=1\).
• **Inductive step**: Assume the statement holds for \(n=k\), and then prove it for \(n=k+1\).

Using this process, once the base case is shown to be true, and the inductive step is validated, the principle of mathematical induction concludes the statement is true for all integers.
Although we do not directly apply induction in our binomial series expansion problem, understanding its principles is crucial. This method can help confirm the validity and generalizability of the steps we take when developing or deriving series expansions.