Maclaurin series is a useful tool to compute nonnegative integer power variables. Here the series having a
function f(x) and order up to n. This uses series Coefficient (f[x, 0, n]) and is inverse Z-transform. This can be
understood and easily evaluated by the following given below examples:-
Example 1: Find the Maclaurin series of the function f(x) = sin x.
Solution 1:
Since the function f(x) = sin x, so f’(x) = cos x, and f”(x) = -sin x, f’”(x) = -cos x, f’”’(x) = sinx. Now the 4th
derivative gets us back to the initial function
Values for these functions if x = 0 will be 0, 1, 0,-1 and again 0, 1, 0,-1 and so on.
Let substitute the values into Maclaurin Series we get
f(x) = sin x = 0+x+0+(-x3)+0+x5+0 ….
Example 2: Find the Maclaurin series of the function f(x) = ex
Solution 2:
Since the function f(x) = ex , so f’(x) = ex, and f”(x) = ex. Now all derivative gets us back to the initial function
f(x) = ex
Values for these functions if x = 0 will be 1,1,1,1, so on.
Let’s substitute the values into Maclaurin Series we get
f(x) = ex = 1 + x + x2/2! + x3/3! ….