Subject: Calculus
Maclaurin Series

Colin Maclaurin (above) is a famous mathematician known for his work in Maclaurin series.
Taylor series provides an approximate for a certain function f(x) at some value x = c. A particular case where c = 0 is called the Maclaurin series, named after a Scottish mathematician, Colin Maclaurin.
The Taylor series in the form
Becomes;
where f’(0) is the first derivative evaluated at x=0, f’’(0) is the second derivative, and so on. This is now the Maclaurin series.
Illustration #1
Find the Maclaurin Series expansion for the following function
Solution:
We first list the derivatives of f(x) evaluated at x=0.
therefore f(0)=0.
so f'(0)=1.
so f''(0)=0.
so f'''(0)=-1.
so f''''(0)=0.
We can observe that the pattern will continue forever. We then substitute these values to the Maclaurin series:
Which then becomes;
Thus, the Mclaurin expansion for the function \sin{x} is
Illustration #2
Find the Maclaurin Series expansion for the function
Solution: We first list the derivatives of f(x) evaluated at x=0.
therefore f(0)=1.
so f'(0)=0.
so f''(0)=-1.
so f'''(0)=-0.
so f''''(0)=1.
We can observe that the pattern is just repeated. We then substitute these values to the Maclaurin series:
Which then becomes;
Thus, the Mclaurin expansion for the function \cos{x} is
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