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Maclaurin Series Formula

A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function. It is a special case of Taylor series when x = 0. The Maclaurin series is given by
Maclaurin SeriesThe Maclaurin series formula is
Maclaurin Series Formula
where f(xo), f'(xo), f''(xo)....... are the successive differentials when xo = 0.

Related Calculators
Maclaurin Series Calculator Series Calculator
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Maclaurin Series Examples

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Lets see some examples for Maclaurin series examples:

Solved Examples

Question 1: Solve f(x) = (x - 3)3 using maclaurin series ?
Solution:
 
Given: Function is f(xo) = (x - 3)3
To find maclaurin series we need to find the derivatives at xo = 0
f(xo) = (x - 3)3
f'(xo) = 3(x - 3)2
f''(xo) = 6 (x - 3)
f'''(xo) = 6
f''''(xo) = 0

At xo = 0,
f(0) = -27
f'(0) = 27
f''(0) = -18
f'''(0) = 6
f''''(0) = 0

The maclaurin series is given by
f(x) = f(xo) + f'(xo)(x - xo) + $\frac{f''(xo)}{2!}$ (x - xo)2 + $\frac{f'''(xo)}{3!}$ (x - xo)3 + .......
     = -27 + 27(x) + $\frac{-18}{2!}$ (x)2 + $\frac{6}{3!}$ (x)3 + ....
     = -27 + 27x + 9x2 + x3 + 0
     = x3 + 9x2 + 27x - 27.
 

Question 2: Solve f(x) = tan x using maclaurin series ?
Solution:
 
Given: Function is f(x) = tan x
To find maclaurin series we need to find the derivatives at xo = 0
f(xo) = tan x
f'(xo) = sec2 x
f''(xo) = 2 sec x (sec x tanx) = 2 sec2x tanx
f'''(xo) = 2 tan x (2sec x) secx tanx + 2 sec2x (sec2 x)
        = 2 sec2x tan2x + 2 sec4x

At x0 = 0,

f(0) = 0
f'(0) = 1
f''(0) = 2(1)(0) = 0
f'''(0) = 2

The maclaurin series is given by
f(x) = f(xo) + f'(xo)(x - xo) + $\frac{f''(x_o)}{2!}$ (x - xo)2 + $\frac{f'''(x_o)}{3!}$ (x - xo)3 + .......
       = 0 + 1(x) + 0 + 2x ....
       = x + 2x + .............

 

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