Problem 23 - Precalculus: Mathematical for Calculus

Question

Use the Binomial Theorcm to expand the expression. $$\left(1+\frac{1}{x}\right)^{6}$$

Step-by-Step Solution

Answer

The expanded expression is \(1 + \frac{6}{x} + \frac{15}{x^2} + \frac{20}{x^3} + \frac{15}{x^4} + \frac{6}{x^5} + \frac{1}{x^6}\).

Step 1: Understand Binomial Expansion Formula

The Binomial Theorem provides a way to expand expressions of the form \((a+b)^n\). Specifically, it states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient. Here, \(a = 1\), \(b = \frac{1}{x}\), and \(n = 6\).

Step 2: Calculate Binomial Coefficients

The binomial coefficients are calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). We'll calculate these for each term in the expansion: - \(\binom{6}{0} = 1\),- \(\binom{6}{1} = 6\), - \(\binom{6}{2} = 15\), - \(\binom{6}{3} = 20\), - \(\binom{6}{4} = 15\), - \(\binom{6}{5} = 6\), - \(\binom{6}{6} = 1\).

Step 3: Expand the Expression Using Binomial Coefficients

Using the values of \(a = 1\) and \(b = \frac{1}{x}\) and the binomial coefficients, expand the expression. The terms are:- \(\binom{6}{0}(1)^6\left(\frac{1}{x}\right)^0 = 1\),- \(\binom{6}{1}(1)^5\left(\frac{1}{x}\right)^1 = \frac{6}{x}\),- \(\binom{6}{2}(1)^4\left(\frac{1}{x}\right)^2 = \frac{15}{x^2}\),- \(\binom{6}{3}(1)^3\left(\frac{1}{x}\right)^3 = \frac{20}{x^3}\),- \(\binom{6}{4}(1)^2\left(\frac{1}{x}\right)^4 = \frac{15}{x^4}\),- \(\binom{6}{5}(1)^1\left(\frac{1}{x}\right)^5 = \frac{6}{x^5}\),- \(\binom{6}{6}(1)^0\left(\frac{1}{x}\right)^6 = \frac{1}{x^6}\).

Step 4: Combine the Terms

Add all the terms calculated in the previous step to get the expanded expression:\[ 1 + \frac{6}{x} + \frac{15}{x^2} + \frac{20}{x^3} + \frac{15}{x^4} + \frac{6}{x^5} + \frac{1}{x^6} \].

Key Concepts

Binomial Expansion

The binomial expansion is a way to express the power of a binomial, which is an expression composed of two terms, into a series of terms using the Binomial Theorem.The theorem offers a method to take any expression in the form \((a+b)^n\) and expand it into a sum of individual terms.Each term in the expansion consists of coefficients, powers of one of the terms, and decreasing powers of the second term from the binomial.
For example, an expression \( (1 + \frac{1}{x})^6 \) becomes a series of terms calculated using the formula:

• \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

Here, the task is to expand the given binomial expression into a detailed series of terms, showing each step.The expansion provides a more manageable way to understand larger powers of binomials by breaking them down into simpler parts.

Binomial Coefficient

Binomial coefficients are crucial components in the binomial expansion.They dictate the weight of each term in the expansion.These coefficients can be found using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \(n!\) denotes the factorial of \(n\), and \(k\) represents the term's position in the expansion.
For instance, when expanding \((1 + \frac{1}{x})^6\), the coefficients will be calculated for each term using this formula:

• \(\binom{6}{0} = 1\)
• \(\binom{6}{1} = 6\)
• \(\binom{6}{2} = 15\)
• \(\binom{6}{3} = 20\)
• \(\binom{6}{4} = 15\)
• \(\binom{6}{5} = 6\)
• \(\binom{6}{6} = 1\)

These coefficients affect the size of each term when the expansion is completed, influencing the final expanded series.

Factorials

Factorials are a key mathematical operation often encountered when working with binomial coefficients.They are denoted by an exclamation mark \(!\), and represent the product of all positive integers up to a specified number.In other words, the factorial of \(n\), written as \(n!\), is calculated as:

• \(n! = n \times (n-1) \times (n-2) \times … \times 1\)

Using this operation is essential for finding the binomial coefficients, as seen in formulas like \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \).Here is a small example of how factorials come into play:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Factorials grow very fast compared to exponential growth, hence indispensable in the calculation of binomial coefficients.