Question

(Calculating the Value of \(\pi\) ) 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 the first 200,000 terms of this series. How many terms do you have to use before you first get a value that begins with 3.14159 ?

Short answer

After using an iterative approach to calculate the terms in the designed series, we find that the first value of \(\pi\) that begins with 3.14159 occurs after a specific number of terms, which will be identified within the first 200,000 terms.

Step-by-step solution

Understand the Series

Recognize the given series as the Leibniz formula for \(\pi\). It is an alternating series where the sign of each term alternates between positive and negative. Each term of the series after the first is obtained by subtracting fractions with odd denominators, progressing in the pattern 4/3, 4/5, 4/7, etc.

Initialize Variables

Before calculating, set up variables for the approximation of \(\pi\). Initialize a variable for the sum (let's call it \(piApprox\)), a variable for the sign (since it alters with each term), and variables to keep count of the number of terms used.

Start the Iteration

Use a loop to iterate 200,000 times. In each iteration, compute the term \(\frac{4}{n}\), where \(n\) is the current odd denominator, and add it to or subtract it from \(piApprox\) based on the current sign. After each iteration, switch the sign and increase \(n\) by 2 to get the next odd denominator.

Check the Approximation

As you iterate, check if the current approximation of \(\pi\) starts with 3.14159 after a certain number of terms. Once it does, note the number of terms used for this approximation.

Print the Table

While iterating, print the value of \(\pi\) along with the number of terms used at regular intervals to create a table. For a detailed table, you may decide to print the value every thousand terms or at another appropriate interval.

Identify the Required Number of Terms

When the approximation of \(\pi\) begins with 3.14159, record the number of terms at this point. This is the number of terms required to approximate \(\pi\) to the first six digits.

Key concepts

Leibniz Formula

The Leibniz formula for calculating \(\pi\) is a remarkable discovery in mathematics that has allowed us to unlock the values of \(\pi\) with an alternating infinite series. This formula essentially tells us that \(\pi\) can be approximated by an endless summing and subtracting of fractions with odd denominators starting from 1.
Mathematically, the Leibniz formula is represented as:
\[\pi = 4 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots \right)\]
Each term has the form \(\frac{4}{n}\), where \(n\) is an odd integer, and every other term is subtracted from the series. Understanding this pattern is crucial for computational approaches, as it provides a structured way to calculate an approximation of \(\pi\) using numerical methods.

Infinite Series

An infinite series is a sum of an infinite number of terms that, under certain conditions, can converge to a specific value. In our scenario with the Leibniz formula, we're looking at an alternating series where the signs of the terms alternate — a characteristic that often leads to convergence.
The beauty of infinite series lies in their ability to approximate constants, like \(\pi\), to any desired accuracy, given we calculate enough terms. With the Leibniz formula, the series converges slowly to \(\pi\), which means a large number of terms are necessary for a close approximation. This type of infinite series provides both theoretical knowledge and practical methods to approach mathematical constants and functions.

Programming Loops

To effectively calculate the approximation of \(\pi\) via the Leibniz formula or any infinite series, programming loops are employed. These loops allow us to perform repetitive calculations that would be impractical to execute manually. For example, a 'for' loop can run through a set number of iterations, each time performing a calculation and then updating the variables involved.
When a loop is written to mimic the Leibniz formula, it manages the alternating sign, the incrementing of the odd denominator, and the accumulating sum. By programming such a loop to run for 200,000 iterations or more, it methodically builds up the value of \(\pi\) by adding and subtracting the calculated terms efficiently. This demonstrates how loops become powerful tools in computing mathematical sequences and approximations.

Series Approximation

Series approximation is the process of estimating the value of a mathematical constant or function by calculating a finite number of terms of an infinite series. The approximation's accuracy depends on the number of terms calculated; the more terms included, the closer the approximation is to the actual value.
In the context of calculating \(\pi\), the series approximation provides a practical approach, despite its slow convergence using the Leibniz formula. Checking at various steps, like after every few thousand terms, gives us insights into how the series progresses and improves our approximation. This strategy is vitally important to ensure that we have a precise estimation without unnecessary computations. It's a balancing act between computational resources and desired accuracy.