Question

Calculate the value of \(\mathrm{p}\) 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 approximate value of \(\mathrm{p}\) after each of the first 1,000 terms of this series.

Short answer

The Gregory-Leibniz series estimates \\ \pi \\ by alternating terms; sum 1000 terms for the approximation.

Step-by-step solution

Understand the Series

The series given is an alternating series that approximates the value of \( \pi \). It's known as the Gregory-Leibniz series: \(\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \cdots\). Each term after the first is an alternation of subtraction and addition.

Identify the General Term

The general term of this series is determined using the formula \( T_n = 4 \left( -1 \right)^{n+1} \left( \frac{1}{2n-1} \right) \). This formula calculates each individual term for the series, where \( n \) is the term number.

Accumulate the Sum

To calculate \( \pi \) as it approximates through the series, sum the individual terms for each \( n \). Begin with \( n=1 \) and continue to \( n=1000 \). If \( S_n \) is the partial sum after \( n \) terms, start with \( S_1 = 4 \) and add each term successively.

Calculate and Display Series

Use a loop or iterative process to calculate the sum for each of the first 1000 terms. Record the value \( S = S + T_n \) for each step. Print or log the values of \( S \) after each term is added.

Key concepts

Gregory-Leibniz series

The Gregory-Leibniz series is a mathematical sequence used to approximate the value of the famous constant, \( \pi \). Formally, it is expressed as:\[\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \cdots\]This series is known for its simple pattern: each term involves back-and-forth subtraction and addition. While the pattern is straightforward, the series converges quite slowly to \( \pi \). Thus, it requires many terms to obtain a precise approximation of \( \pi \). Understanding this series helps in grasping how infinite sequences can represent real numbers when extended indefinitely.
Key details include:

• It is an alternating series, highlighted by the alternation of positive and negative terms.
• The denominators are odd numbers that increase sequentially.
• Each term follows the multiplication pattern of \(4\) with the sequence \(\frac{(-1)^{n+1}}{2n-1}\).

Approximation of Pi

Approximating \( \pi \) using the Gregory-Leibniz series is an insightful exercise that exemplifies how infinite series can provide practical approximations of mathematical constants. In this particular series, finite sums are used to approach the exact value of \( \pi \) gradually. When computing the sum for multiple terms, the resulting value gets closer to \( \pi \).The slow convergence of the series means it takes thousands of terms to achieve significant accuracy. For instance, adding only a couple hundred terms might yield an estimation of \( \pi \) that has only a few decimal places of precision. However, adding more terms, such as 1,000 or more, increases the number of accurate decimal places.
This method of approximation is significant because it demonstrates a clear use case of infinite series in real-world computations, specifically in the discipline of numerical methods and calculus.

Alternating Series

An alternating series is a series of numbers in which the signs of the terms alternate between positive and negative. The Gregory-Leibniz series is a perfect example:\[4, -\frac{4}{3}, \frac{4}{5}, -\frac{4}{7}, \cdots\]Such series are important in calculus due to their specific convergence properties. In an alternating series, terms gradually approach zero, contributing to the convergence of the entire series. This gradual change in terms helps stabilize the approximation of a value, like \( \pi \) in our case.
Some features of alternating series are:

• The alternating sign changes ensure that the series doesn't diverge easily, guiding it towards a limit.
• The absolute value of terms usually decreases steadily, contributing to the balance and eventual convergence of the series.
• The convergence conditions, like the Alternating Series Test, provide a way to determine if the series really approaches a definite limit.

Series Convergence

Series convergence refers to the behavior of an infinite series as the number of terms increases. If the sum of the series approaches a finite limit, the series is said to converge. In the context of the Gregory-Leibniz series, we speak of convergence towards the value of \( \pi \). Each partial sum, \( S_n \), becomes a better approximation of \( \pi \) as \( n \) increases.The success of using series convergence lies in mathematical techniques that test and assure convergence, ensuring that calculations using infinite series yield meaningful results. For the Gregory-Leibniz series:

• As more terms are included, the series heads towards the true value, \( \pi \).
• Despite its slow convergence, each additional term drives the sum closer to the target value.
• The convergence can be observed numerically by computing the partial sums and analyzing their values.

These concepts of convergence are not only fundamental in calculus but are also critical in understanding why infinite series have real applications in mathematical problem-solving and approximations.