Sine Series Finite Sum - C PROGRAM

In the last post, I discussed about how would one go about calculating the sum of a finite series using C.

In this post I will apply that method, to find the sum of the Sine series for only a finite number of terms.

Sine series is given by:
$\sin (x) = x-\frac{x^3}{3!} + \frac{x^5}{5!} - ....$

We will start the numbering the terms from 0. That is, $t_0=x$ , $t_1=-\frac{x^3}{3!}$ , ….

It’s easy to see that the ratio between consecutive terms is given by:

$R=\frac{t_i}{t_{i-1}}=-\frac{x^2}{(2i+1)2i}$

Since, we indexed the terms starting from 0, therefore, for the above relation to work, $i$ will go from 1 to $n$ .

[Hint: To find the general form of the ratio given in the above expression, try writing down t1/t0, t2/t1,…and then you would be able to see the ratio.]

Now, knowing the first( $t_0$ ) term, the successive terms can be calculated as :

$t_1=R\times t_0$

$t_2=R\times t_1$

and so on.

Therfore, the C program that calculates the sum of the sin series upto a given number of terms can be written as shown below.

PROGRAM:

/********************************
******FINITE SERIES SUM**********
Series: sin(x) = x - (x^3/3!) + (x^5/5!) + ..... 
********************************/
#include<stdio.h>
#include<math.h>
main(){
	int i,n;
	double x,t0,t1,R,sum;
	printf("Enter the value of x:\n");
	scanf("%lf",&x);
	printf("Enter the no. of terms to be summed: ");
	scanf("%d",&n);
	//Initialize First Term
	t0=x;
	//Make sum equal to the first term
	sum=t0;
	printf("n\ttn\t\tSn\n_________________________________");
	for(i=1;i<n;i++){
		//Find the ratio of the second term to the first term using already known relation
		R=-(x*x)/(2*i+1)/(2*i);
		//Calculate the second term
		t1=R*t0;
		//find the new sum
		sum=sum+t1;
		t0=t1;
		printf("\n%d\t%f\t%lf\n",i+1,t1,sum);
	}
	printf("\nThe sum is: %f",sum);
}

The program also prints the value of each term(except the first( $t_0$ ) term) and sum(partial) upto that term.

OUTPUT:

The output of the above program for various values of $x$ and no. of terms is shown below:

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Manas Sharma

Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.

manas.bragitoff.com/