(Compute π) You can approximate πby using the following series: π=√(6 ×(1+(1)/(4)+(1)/(9)+(1)/(1

(Compute π) You can approximate πby using the following...

(Compute $\pi$ ) You can approximate $\pi$ by using the following series:
$$
\pi=\sqrt{6 \times\left(1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\ldots+\frac{1}{n^2}\right)}
$$

Write a program that displays the $\pi$ value for $n=100,200, \ldots$, and 600 .

Key Concepts

Infinite Series

An infinite series is the sum of an endless sequence of terms. In mathematics, certain infinite series converge to finite values, allowing us to approximate constants or functions by summing a manageable number of terms.

Basel Problem

The Basel problem is a famous historical problem that determined the sum of the reciprocals of the squares of the natural numbers equals ?²/6. This result underpins the series used for approximating ? in the problem.

Partial Sum Approximation

Partial sum approximation involves summing a finite number of terms from an infinite series to estimate the total sum. As more terms are included, the approximation becomes closer to the actual value, illustrating the concept of convergence.

Algorithm Implementation

Implementing the series in a programming context requires iterative loops to accumulate the sum and the use of functions to perform calculations such as square roots. This demonstrates how mathematical series are translated into computational algorithms for numerical approximation.

Transcript

00:01 Okay, we have an equation for the value of pi and we want to try it for a whole bunch of terms, everything from 10 ,000 terms all the way up to 100 ,000 terms in increments of 10 ,000.

00:16 And so let's calculate the stuff in parentheses first.

00:21 And i'm just going to call that answer.

00:23 It's not the total answer that pi is.

00:26 And i'm going to call that 0 .0 because i want that to be a floating point value.

00:30 You and then i'm going to build a first four loop and i'm just going to use it value and i'm going to say in range and i'm going to start off with 10 ,000 because that's where i want to start i want to go to a hundred thousand so i'll write a hundred thousand and one and then i want to increment by ten thousands and so i suppose you could do a hundred thousand and fifty if you want it doesn't have to be 100 ,0001, but that's the smallest value that you need there.

01:13 And then we're going to be using an equation with a variable in it.

01:22 And so i'm going to say another four loop for i'll call it i in range.

01:34 And i'm going to go from one to value.

01:38 And so right here i'm to say value plus one so that we end up at value.

01:45 So if value's 10 ,000, that's going to give me 1 to 10 ,000.

01:50 And then what i'm going to do is i need to use floating point numbers in order to get this to work.

01:55 So i'm just going to convert that i, that integer i to a floating point and call it x and use the x in the equation.

02:07 And so now i'm going to do answer.

02:09 And i'm going to increment.

02:11 Because we're summing over all of those terms.

02:16 And the first piece i want is i'm going to do minus 1 raised to the x plus 1 power.

02:32 And that's what that part is right there.