Problem 63 - Thomas Calculus

Question

Estimating Pi The English mathematician Wallis discovered the formula \begin{equation}\frac{\pi}{4}=\frac{2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdot 8 \cdot \cdots}{3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot \cdots}\end{equation} Find \(\pi\) to two decimal places with this formula.

Step-by-Step Solution

Answer

Pi estimated to two decimal places is 3.14.

Step 1: Understand Wallis' Formula

Wallis' formula can be expressed as \(\frac{\pi}{4} = \prod_{n=1}^{\infty} \frac{(2n)(2n)}{(2n-1)(2n+1)}\). To estimate \(\pi\) using this formula, we approximate the infinite product by multiplying a finite number of terms.

Step 2: Decide the Number of Terms to Calculate

Choosing the number of terms is key to getting an accurate approximation. Since we need \(\pi\) to two decimal places, we will calculate the product for 10 terms (\(n=1\) to \(n=10\)). This will provide a practical balance between complexity and accuracy.

Step 3: Calculate the Product for 10 Terms

Calculate each term of the product \(\frac{(2n)(2n)}{(2n-1)(2n+1)}\) from \(n=1\) to \(n=10\) and multiply them together. The terms are: \(\frac{2dot 2}{1 dot 3}\), \(\frac{4dot 4}{3 dot 5}\), \(\frac{6dot 6}{5 dot 7}\), ..., \(\frac{20 dot 20}{19 dot 21}\). Do these calculations one by one and keep multiplying them to get the approximate value of \(\frac{\pi}{4}\).

Step 4: Multiply the Product by 4

Once the approximate product \(P\) is computed, calculate \(\pi\) by using \(\pi = 4P\). Suppose \(P\) was found to be approximately 0.785398163, multiply this by 4 to get a \(\pi\) approximation of 3.141592653.

Step 5: Round to Two Decimal Places

Round the calculated value of \(\pi\) to two decimal places. Given that our product yielded an approximation for \(\pi\) of 3.141592653, rounding results in \(\pi \approx 3.14\).

Key Concepts

Pi Approximation

Wallis' formula allows us to estimate the value of \(\pi\) through a sequence of fractions, known as an infinite product. This method is particularly important because \(\pi\) is an irrational number, meaning it cannot be expressed as a simple fraction. Thus, we must use special techniques to approximate it. Wallis' infinite product converges towards \(\pi\) slowly, but its simplicity makes it a favorite among mathematicians.

In practical terms, when approximating \(\pi\) using Wallis' formula, rather than calculating infinite terms, we choose a manageable number of terms. This approach strikes a balance between accuracy and computational complexity. In this instance, calculating ten terms provides a sufficient approximation for most practical applications.

Remember, the more terms you compute, the closer you get to \(\pi\). But for general purposes, just like our exercise aimed for, a two-decimal-place accuracy (\