Maclaurin Series Basics

AHL 5.19

Maclaurin series allow us to approximate arbitrary functions as polynomials.

The Maclaurin series for a function f is given by

f(x)=f(0)+xf′(0)+2!x2f′′(0)+⋯📖

In summation form:

f(x)=n=0∑∞n!f(n)(0)xn🚫

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AHL 5.19

The Maclaurin series for ex is

ex=1+x+2!x2+⋯📖

AHL 5.19

The Maclaurin series for sinx is

sinx=x−3!x3+5!x5−⋯📖

AHL 5.19

The Maclaurin series for cosx is

cosx=1−2!x2+4!x4−⋯📖

AHL 5.19

The Maclaurin series for ln(x+1) is

ln(x+1)=x−2x2+3x3−⋯📖

AHL 5.19

The Maclaurin series for arctanx is

arctanx=x−3x3+5x5−⋯📖

AHL AA 1.10

The binomial theorem can be extended to expansions with rational exponents (n∈Q):

(a+b)n=an(1+n(ab)+2!n(n−1)(ab)2+⋯)📖

Exercises