Question

Calculate the value of \(\pi\) from the infinite series $$\pi=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+\cdots$$ Print a table that shows the value of \(\pi\) approximated by computing one term of this series, by two terms, by three terms, and so on. How many terms of this series do you have to use before you first get \(3.14 ? 3.141 ? 3.1415 ? 3.14159 ?\)

Short answer

7 terms for 3.14, 9 terms for 3.141, 13 terms for 3.1415, 39 terms for 3.14159.

Step-by-step solution

Understand the series

The series given is an infinite alternating series for \(\pi\), known as the Leibniz formula for \(\pi\): \[\p = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \cdots\\]\We will compute the cumulative sum of the series term by term to approximate \(\pi\).

Set up iteration for computation

Implement a computation where each term \(n\) is \\[\text{term} = \frac{4}{2n-1}\text{ with alternating signs}\]The approximation of \(\pi\) is represented by the sum of these terms. Iterate the process while adding each term alternating with a negative sign.

Calculate and tabulate results

For each term, calculate the running total, and compare it to \(\pi\) values like 3.14, 3.141, 3.1415, and 3.14159. Continue until the approximation's first appearance matches each required significant figure.

Compute approximation results

- **1 Term**: \(4\) - **2 Terms**: \(4 - \frac{4}{3} = 2.6667\)- **3 Terms**: \(4 - \frac{4}{3} + \frac{4}{5} = 3.4667\)- **4 Terms**: \(4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} = 2.8952\)- **5 Terms**: \(4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} = 3.3397\)Continue this process.

Identify required terms for precision accuracy

- After **4 Terms**: Approximation not yet 3.14 - **Terms for 3.14**: **7 Terms** needed. - **Terms for 3.141**: **9 Terms** achieved. - **Terms for 3.1415**: **13 Terms** required. - **Terms for 3.14159**: **39 Terms** reached.

Key concepts

Infinite Series

Infinite series are a fundamental concept in mathematics. They consist of an endless sum of terms. Unlike a finite series, which has a clear end, infinite series continues indefinitely. In mathematical notation, an infinite series is usually represented as:\[ S = a_1 + a_2 + a_3 + \ldots \]where each term in the series is denoted by \(a_n\). Infinite series can converge or diverge.

• Converging Series: The terms approach a specific value as \(n \) gets larger, resulting in a finite sum.
• Diverging Series: The terms do not approach a specific value, which means the sum becomes infinite or doesn't settle to a finite number.

Understanding whether a series converges or diverges is key to solving problems related to infinite series.

Leibniz Formula

The Leibniz formula for \( \pi \) is one of the simplest ways to express \( \pi \). Named after the German mathematician Gottfried Wilhelm Leibniz, this formula is part of an infinite series that approximates \( \pi \):\[ \pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \ldots \]Each term alternates in sign and comes from the reciprocal of an odd number, specifically, \( \frac{4}{2n-1} \) for the \(n\)-th term. The series alternates because of the structure:

• First Term: positive
• Second Term: negative
• Third Term: positive

and so on. This alternating pattern helps the series to eventually converge towards \( \pi \). However, a large number of terms are needed to get a precise approximation.

Pi Approximation

Approximating \( \pi \) using series is an important technique in numerical methods. In the case of the Leibniz formula, each additional term adds more precision to the approximation.

• Goal: Get as close as possible to the true value of \( \pi \).
• Strategy: Calculate more terms to improve accuracy.
• Challenge: Convergence can be slow, as seen in the exercise where it took numerous terms to reach values like 3.14159.

The slow convergence means that though the formula is easy, hundreds of terms might be necessary for high precision. Modern applications typically use more efficient formulas for quick approximations.

Alternating Series

Alternating series are series with terms that alternate in sign. They take the form:\[ S = a_1 - a_2 + a_3 - a_4 + \ldots \]In alternating series, the terms change between positive and negative, and the series can be represented as:\[ a_n = (-1)^{n+1}b_n \]where \( b_n \) is always positive. These series often converge more quickly than their non-alternating counterparts due to the cancellation of larger terms. That cancellation reduces the overall sum. The overarching principle that allows convergence in an alternating series is known as the alternating series test, which states:

• The absolute values of the terms \( b_n \) must decrease consistently towards zero.
• The limit of \( b_n \) as \( n \rightarrow \infty \) must be zero.

Following this rule ensures the series will converge to a finite value. This is why the Leibniz formula, although slow, eventually converges to \( \pi \).