Write a program that approximates the value of πby summing the terms of this series: 4 / 1- 4 /

Write a program that approximates the value of πby summing...

Write a program that approximates the value of $\pi$ by summing the terms of this series: $4 / 1-$ $4 / 3+4 / 5-4 / 7+4 / 9-4 / 11+\ldots$ The program should prompt the user for $n$, the number of terms to sum, and then output the sum of the first $n$ terms of this series. Have your program subtract the approximation from the value of math. pi to see how accurate it is.

Key Concepts

Infinite Series Approximation

This concept involves representing a mathematical constant, such as pi, as the sum of an infinite sequence of terms. By summing a finite number of terms from the series, one can approximate the constant to a desired degree of accuracy. It highlights the idea that infinite processes can be approached through finite calculations.

Alternating Series

An alternating series is a sequence of terms in which the signs alternate between positive and negative. This concept is crucial when summing series like the Leibniz formula for pi, where the cancellation between successive terms influences the convergence properties and the accuracy of the approximation.

Series Convergence and Error Analysis

Series convergence refers to the process by which the sum of the terms approaches a specific value as more terms are added. Error analysis in this context involves comparing the approximation obtained from a finite number of terms with the actual value, allowing for an assessment of the estimation's accuracy and the improvement gained by including additional terms.

Iteration and Looping Constructs in Programming

Iteration using loops is a fundamental programming technique that facilitates the repeated execution of a sequence of instructions. In the context of approximating pi from a series, loops are used to iterate through the required number of terms to compute the partial sum, making the algorithm dynamic and responsive to user input.

Transcript

00:01 So we're going to write a program that approximates the value of pi by summing the terms for this series.

00:06 4 divided by 1, minus 4 divided by 3, plus 4 divided by 5, and so on.

00:12 So we're alternating plus, minus, plus, minus.

00:15 And our numerator is always 4, and our denominator goes from 1, 3, 5, 7, 9, 11.

00:21 We're going to ask the user how many terms they want to use.

00:26 And then we will sum the first n terms of the series for how many they ask.

00:32 Then we're going to use the program subtract our approximation from math .py to see just how accurate it is.

00:39 So here's what i see.

00:41 Plus 4 over 1, minus 4 over 3, plus 4 over 5, and so on.

00:45 So we're going to use some sort of counter, a for loop, to go from 0 up through our n minus 1 term.

00:56 So we have n terms, but we want to start counting at 0 because that's what computers do, 0 indexing.

01:02 So let's say they want six terms we're gonna have 0 1 2 3 4 5 that's six terms how does that work with our denominator well i see this pattern 2i plus 1 2 times 0 plus 1 is 1 2 times 1 plus 1 is 3 2 times 2 plus 1 is 5 and so on if i is odd we're gonna subtract if i is even we're gonna add and we'll use remainder division to see if we have an odd or an even index up here so counting index so so let's go ahead and see what the program looks like.

01:35 So we're going to use math .py.

01:36 So i'm going to import math.

01:38 All my imports should happen at the top of the program.

01:41 I'm going to get some input.

01:43 I'm going to ask the user, how many terms do you want to sum? i'm going to cast that to be an integer.

01:49 And i'm going to assign it to a variable named number of terms.

01:54 I'm going to set up my approximate pi variable, initialize it to zero.

02:00 And i'll be adding to it as we go along.

02:02 So based on, and we're going to, we don't have any error checking here.

02:05 We're going to assume that our user is smart enough to know that they have to enter a number for this.

02:09 If they don't, we'll get an error.

02:11 It doesn't seem like the textbook has gotten to error checking yet...