Question

Use a graphing utility to show that \(\frac{\sqrt{8}}{9801} \sum_{n=0}^{\infty} \frac{(4 n) !(1103+26,390 n)}{(n !) 396^{4 n}}=\frac{1}{\pi}\) (Note: This series was discovered by the Indian mathematician Srinivasa Ramanujan in \(1914 .)\)

Short answer

By following these steps, one would be able to graphically verify that the given series equals the reciprocal of Pi using a graphing utility. However, the series can only be graphed up to a certain number of terms since it is infinite. The verification comes from the observation of the overlapping or very close approximation of the respective graphs as \(n\) increases.

Step-by-step solution

Identify the series

The given series is: \[ \frac{\sqrt{8}}{9801} \sum_{n=0}^{\infty}\frac{(4 n) !(1103+26,390 n)}{(n !) 396^{4 n}} \] Our goal is to show that this series is equal to \(\frac{1}{\pi}\).

Simplify the series expression

Before graphing, it's essential to express the series in a more simplified form. For each term in the series: \((4n)! \) is the factorial of \(4n\), \( (n!) \) is the factorial of \(n\), and \( 396^{4n} \) means \(396\) raised to the power of \(4n\). The term \(1103+26,390n\) is a simple linear function. Use these explanations to simplify the series.

Use a graphing utility

Enter the simplified series into a graphing utility. It's suggested to graph the series within its convergence radius. Let \(x\) axis represent the \(n\) and the \(y\) axis represent the series expression. Graph the series according to each term. Note that because the series is infinite, you will only be able to graph a finite number of terms, but the pattern should become evident.

Graph the reciprocal of Pi

Next graph the function \(y = \frac{1}{\pi}\) in the same window. This will allow you to visually compare the series with \(\frac{1}{\pi}\). If the series is indeed equal to \(\frac{1}{\pi}\), the graphs should overlap, or at least become increasingly close as \(n\) grows larger.

Key concepts

Graphing Utility

A graphing utility is an essential tool in mathematics, particularly when dealing with complex concepts such as the Ramanujan series for 1/π. With this technology, students can visualize infinite series and better understand their behavior. By plotting the partial sums of the series against the corresponding terms, students can observe whether the sequence of partial sums appears to stabilize or diverge as the number of terms increases.

Using a graphing utility, after inputting the simplified expression of the Ramanujan series, one can graph the terms against the natural numbers to see the progression of the series visually. Overlapping the graph of the reciprocal of π with the series as suggested in step 3 and 4 of the provided solution establishes a compelling visual understanding of their equivalence. For exercises involving proving the series equals 1/π, visually demonstrating this through graphing utilities greatly enhances comprehension. The graphing utility serves not just to prove, but to inspire intuition about how the series behaves.

Infinite Series

When we talk about an infinite series, we're referring to the sum of infinitely many terms that follow a certain pattern. This might seem counterintuitive at first—how can we add up an infinite number of things? The answer lies in the concept of a limit. In mathematics, if the terms of a series become smaller and smaller, approaching zero, the series may converge to a finite value.

In the case of the Ramanujan series for 1/π, the series contains a vast number of terms that, when summed up from n=0 to infinity, approach the actual value of 1/π. The elegance of the series is that each additional term brings the partial sum closer to 1/π. To understand an infinite series like Ramanujan’s, comprehending that we're looking at the limit of the partial sums as the number of terms goes to infinity helps ground the concept in something more tangible.

Factorials in Series

The factorial, denoted by an exclamation mark (!), is a function that multiplies a number by all the positive integers below it. For example, 5! is 5 x 4 x 3 x 2 x 1. In the context of series, factorials can add significant complexity due to their rapid growth rate. As illustrated in the Ramanujan series, we encounter terms like (4n)! and (n!), which can become astronomically large even for relatively small values of n.

Understanding factorials in series is critical when dealing with convergence, as their immense growth can affect the series' behavior. However, in the Ramanujan series, despite the presence of large factorials, the series converges due to the balance with other terms in the expression, specifically the powers of 396 to the fourth power of n, which stifles the growth enough for convergence to be achieved.

Mathematical Convergence

The convergence of an infinite series is a testament to the beauty of mathematics—where infinity meets the finite. Convergence refers to the behavior of a series such that as more terms are added, the sum approaches a specific finite value. This contrasts with divergence, where adding more terms causes the sum to grow without bound or oscillate without approaching any value.

For the Ramanujan series, mathematical convergence is exemplified through the intricate balance of the series terms. The factorial in the numerator is tempered by the exponential term in the denominator, ensuring that each additional term has diminishing impact on the overall sum, letting it converge to a finite number—specifically, 1/π. Students should appreciate that convergence is not about the size of individual terms, but about the series’ behavior as more and more terms are summed up. It’s a dance between the growth of factorials and the moderating influence of powers, which in the end, gracefully concludes at a specific, known value.