In my recent posts I have showed you how to write C programs that calculate the sum of a finite as well as an infinite series.

Now, that we have discussed a few examples like the Sine, Cosine and the Bessel series, we can try some harder problems.

In this post we will discuss three more series.

### Q. Evaluate given by

correct to four decimal places, for , at intervals of 0.1.

**Solution.**

The first term of the series is:

and the ratio of the consecutive terms is:

### PROGRAM:

/***************************** ******SERIES PROBLEM********* ****************************/ #include<stdio.h> #include<math.h> main(){ FILE *fp=NULL; fp=fopen("seriesProblem4.txt","w"); double t0,t1,R,sum,z,eps; printf("Enter the desired accuracy: "); scanf("%lf",&eps); for(z=0;z<=1;z=z+0.1){ int i=1; //Initialize First Term t0=z; //Make sum equal to the first term sum=t0; do{ //Find the ratio of the second term to the first term using already known relation R=-(M_PI*M_PI*pow(z,4))/((4*i+1)); //Calculate the second term t1=R*t0; //find the new sum sum=sum+t1; t0=t1; i++; //keep on summing terms until the required accuracy is reached }while(fabs(t1/sum)>eps); sum=sum*cos(M_PI*z*z/2); fprintf(fp,"%lf\t%lf\n",z,sum); } }

### OUTPUT:

When you run the above program, it will just ask you to enter the desired accuracy, create a file called `'seriesProblem4.txt'`

containing the data-points in the given range of x. You can then plot this file using GnuPlot by giving the command:

`->plot 'seriesProblem4.txt' w l`

The output looks something like this:

### Q. Write a program to plot the sum of the following series:

for and in the range . You would require the following relations:

**Solution.**

The first term of the series is:

and the ratio of the consecutive terms is:

### PROGRAM:

/***************************** ******SERIES PROBLEM********* ****************************/ #include<stdio.h> #include<math.h> main(){ FILE *fp=NULL; fp=fopen("seriesProblem5.txt","w"); double t0,t1,R,sum,z,eps; printf("Enter the desired accuracy: "); scanf("%lf",&eps); for(z=0;z<=5;z=z+0.01){ int k=2; //Initialize First Term t0=1/(2*sqrt(M_PI)); //Make sum equal to the first term sum=t0; do{ //Find the ratio of the second term to the first term using already known relation R=(4*z*z*(3-k))/(2*k*(k-1)); //Calculate the second term t1=R*t0; //find the new sum sum=sum+t1; t0=t1; k=k+2; //keep on summing terms until the required accuracy is reached }while(fabs(t1/sum)>eps); fprintf(fp,"%lf\t%lf\n",z,sum); } }

### OUTPUT:

When you run the above program, it will just ask you to enter the desired accuracy, create a file called `'seriesProblem5.txt' `

containing the data-points in the given range of x. You can then plot this file using GnuPlot by giving the command:

`->plot 'seriesProblem5.txt' w l`

The output looks something like this:

### Q. Write a program to plot the following function:

where , for in the range , at intervals of 0.05.

**Solution.**

The first term of the series is:

and the ratio of the consecutive terms is:

### PROGRAM:

/***************************** ******SERIES PROBLEM********* ****************************/ #include<stdio.h> #include<math.h> main(){ FILE *fp=NULL; fp=fopen("seriesProblem6.txt","w"); double t0,t1,R,sum,z,eps; printf("Enter the desired accuracy: "); scanf("%lf",&eps); for(z=-10;z<=0;z=z+0.05){ int i=1; //Initialize First Term t0=1; //Make sum equal to the first term sum=t0; do{ //Find the ratio of the second term to the first term using already known relation R=pow(z,3)/((3*i)*(3*i-1)); //Calculate the second term t1=R*t0; //find the new sum sum=sum+t1; t0=t1; i++; //keep on summing terms until the required accuracy is reached }while(fabs(t1/sum)>eps); sum=sum*0.35503; fprintf(fp,"%lf\t%lf\n",z,sum); } }

### OUTPUT:

When you run the above program, it will just ask you to enter the desired accuracy, create a file called `'seriesProblem6.txt' `

containing the data-points in the given range of x. You can then plot this file using GnuPlot by giving the command:

`->plot 'seriesProblem6.txt' w l`

The output looks something like this:

### Write a program to evaluate the sum up to 20 terms of the series

for a given x, and compare your result with the analytic sum of the series.

**Solution:**

### PROGRAM:

/******************************** ******FINITE SERIES SUM********** Series: S(x) = 1 + (1/x^2) + (1/x^3) + ..... + (1/x^n) ********************************/ #include<stdio.h> #include<math.h> main(){ FILE *fp=NULL; fp=fopen("seriesProblem1.txt","w"); int i,n; printf("Enter the number of terms to be summed(n): "); scanf("%d",&n); double x,xmin,xmax; printf("Enter the the range of x:\nxmin = "); scanf("%lf",&xmin); printf("xmax = "); scanf("%lf",&xmax); for(x=xmin;x<=xmax;x=x+0.1){ /*Initialize t0 with the value of the first term of the series */ double t0=1/(x*x); double t1,R,sum=1+t0; for(i=1;i<n-1;i++){ R=(double)1/x; t1=R*t0; t0=t1; sum=sum+t1; } fprintf(fp,"%lf\t%lf\n",x,sum); } }

### REFERENCES:

The above problems have been taken from the Computer Programming & Numerical Analysis Manual by Dr. Shobhit Mahajan.