Taylor and Maclaurin Series

Lets try with Taylor and Maclaurin Series. We start by supposing that ƒ is any function that can be represented by a power series:

Let’s try to determine what the coefficients c_n must be in terms of ƒ. To begin, notice that if we put x = a in Equation 1, then all terms after the first one are 0 and we get

ƒ(a) = c₀

If we differentiate the series in equation 1 term by term...

Here we just convert the degree in radian as we know that
180⁰ = π^c

And the substitution of x = a in equation 2

ƒ^'(a) = c₁

Now we differentiate both sides of Equation 2 and obtain

Again we put x = a in Equation 3. The result is

ƒ^''(a) = 2c₂

Let’s apply the procedure one more time. Differentiation of the series in Equation 3 gives

ƒ^'''(a) = 2.3c₃ = 3!c₃

By now you can see the pattern. If we continue to differentiate and substitute x = a, we obtain

Solving this equation for the n^th coefficient c_n, we get

This formula remains valid even for n = 0 if we adopt the conventions that 0! = 1 and ƒ⁽⁰⁾ = ƒ.

Thus we have proved the following theorem.
THEOREM: If ƒ has a power series representation (expansion) at a, that is, if

Substituting this formula for c_n back into the series, we see that if ƒ has a power series expansion at a, then it must be of the following form.

This equation is called the Taylor series of the function ƒ at a (or about a or centered at a). For the special case a = 0 the Taylor series becomes

This case arises frequently enough that is is given the special name Maclaurin series.

EXAMPLE 1: Find the Maclaurin series for sin x and prove that it represents sin x for all x.
Solution: We arrange our computation in two columns as follows:

Since the derivatives repeat in a cycle of four, we can write the Maclaurin series as follows:

See the C program to find out sin(x)

EXAMPLE 2: Find the Maclaurin series for cosx and prove that it represents cosx for all x.
Solution: We arrange our computation in two columns as follows:

See the C program to find cos(x)

Collected from Calculus Website, by Kiryl Tsishchanka

C program For Trigonometric ratio.

Lets try with the library function to calculate the trigonometric ratio. Here use the built in function. The C library function double cos(double x) returns the cosine of a radian angle x. Correspondingly the sin(x) and tan(x)

Here we just convert the degree in radian using 180⁰ = π^c (Radian)

// Using Library function
#include<stdio.h>
void main(){
    float tempt, degree;
    printf("Enter the number \(In degree\)\n");
    scanf("%f",&degree);
    tempt = degree;
    degree = degree * 3.14159265359 / 180;
    printf("sin\(%.2f\) = %1.2f\n",tempt, sin(degree));
    printf("cos\(%.2f\) = %1.2f\n",tempt, cos(degree));
    printf("tan\(%.2f\) = %1.2f\n",tempt, tan(degree));


}

Now try with Taylor and Maclaurin Series Following are these equations.. we will transform them in our program.

#include<math.h>
#include<stdio.h>
double  factorial(int num){
        double fact =1;
        int i;
        for(i=1;i<=num;i++){
            fact = fact*i;
        }

        return fact;
}

void main(){
    double tempt, x,sum=1.0;
    int term,counter;
    int  sign =-1.0;
    printf("Enter the term number till you want to calculate the value\n");
    scanf("%d",&term);
    printf("Enter the value of x\n");
    scanf("%lf",&x);
    tempt=x;

//ex
    for(counter=1;counter <= term ; counter++){
        sum = sum + ((pow(x,counter))/(factorial(counter)));
    }
 printf("So the Taylor series of ex around %.1lf is %.15lf \n",tempt,sum);

//sinx
x = x * 3.14159265359 / 180;
sum = x;
for(counter=3;counter<=term;counter+=2){
   sum = sum + (sign * (pow(x,counter))/factorial(counter));
}
printf("The Taylor series of sin(x) around %.1lf is %.15lf \n",tempt,sum);


// cosx
 sum = 1.0;
 sign = -1.0;
 for(counter=2;counter <= term ; counter += 2){
        sum = sum + (sign *(pow(x,counter))/factorial(counter));
        sign = sign * (-1);
    }
 printf("The Taylor series of cos(x) around %.1lf is %.15lf \n",tempt,sum);


}

Explanation

Here we use a user defined function factorial which takes an integer value to calculate it's factorial and the function also return a double type integer.

Here we use some variables
x = the variable of Taylor series.
sum = to hold the value of cos, sin and e^x
tempt = to hold the value of x, as we change the value of x into radian.
term = Number of term of Taylor series.
counter = loop counter.
sign = to make negative number.

---

e^x

We initialize the sum with 1, and the series of e^x also start with 1 then just use it's corresponding arithmetic expression

sin(x)

Again we initialize the sum with 1 since the series of cos(x) start with 1, the counter variable start with 2 and each time it increases by 2, as the series goes. and the sign variable also changes its negativity after each iteration.

cos(x)

Notice that the sin(x) series start with x so again we initialize the sum with x and goes as like the Taylor series.