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

*in Equation 1, then all terms after the ﬁrst one are 0 and we get*

**x = a** ƒ*(a) = c _{0}*

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

Here we just convert the degree in radian as we know that

180^{0} = π^{c}

And the substitution of * x = a* in equation 2

ƒ^{'}*(a) = c _{1}*

Now we diﬀerentiate both sides of Equation 2 and obtain

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

ƒ^{''}*(a) = 2c _{2}*

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

ƒ^{'''}*(a) = 2.3c _{3} = 3!c_{3}*

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

*coeﬃcient c*

**n**^{th}_{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

**for all x.**

*sin 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

**for all x.**

*cosx***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