Maclaurin Series

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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! …. 



 

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