Summary Taylor and Maclaurin Series – Comprehensive Study Notes (PDF)

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Taylor and Maclaurin Series
In the context of power series of the form:
∞

n
∑ Cn × X

n=0




Let's expand our understanding of these series by looking at the power expansion:
∞

n
∑ C n × (X − A)

n=0




Solving for Coefficients
To solve for the coefficients, find

F (A)



. All binomials become zero, except for the first coefficient,

C0



.

F (A) = C 0




Differentiation
Taking the derivative of the function:




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The derivative of any constant is zero, so

C0



goes away.
The term

(X − A)



will go away, leaving

C1



.

Using the chain rule on the other terms:

d 2
[C 2 × (X − A) ] = 2 × C 2 × (X − A)
dx


d 3 2
[C 3 × (X − A) ] = 3 × C 3 × (X − A)
dx


d 4 3
[C 4 × (X − A) ] = 4 × C 4 × (X − A)
dx


Plugging in

A


will make all terms disappear, leaving

C1



.
′
F (A) = C 1



Taking the derivative again:
′′
F (A) = 2 × C 2



Therefore:
′′
F (A)
C2 =
2




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