Maclaurin and Taylor Series Shortcuts
Introduction
Taylor series approximate functions by polynomials around a point a. The Maclaurin series
is a special case centered at a = 0. These expansions are essential for approximations, solving
differential equations, and numerical analysis.
Core Maclaurin Series
∞
X xn x2 x3
ex = =1+x+ + + ···
n=0
n! 2! 3!
∞
X x2n+1 x3 x5
sin x = (−1)n =x− + − ···
n=0
(2n + 1)! 3! 5!
∞
X x2n x2 x4
cos x = (−1)n =1− + − ···
n=0
(2n)! 2! 4!
Shortcuts and Manipulations
• Substitution: Replace x with another expression. Example: ex = ∞ x2n
2 P
n=0 n!
.
(−1)n xn
• Negative Input: e−x = ∞
P
n=0 n!
.
• Integration: Integrate term by term. Example: ex dx = ∞ xn+1
R P
n=0 (n+1)! .
• Differentiation: Differentiate term by term. Example: d
dx
sin x = cos x matches the
series term by term.
P∞ n x4n+2
• Composition: Plug one series into another. Example: sin(x2 ) = n=0 (−1) (2n+1)! .
Applications
• Approximating values of functions near x = 0.
• Solving limits and indeterminate forms.
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Introduction
Taylor series approximate functions by polynomials around a point a. The Maclaurin series
is a special case centered at a = 0. These expansions are essential for approximations, solving
differential equations, and numerical analysis.
Core Maclaurin Series
∞
X xn x2 x3
ex = =1+x+ + + ···
n=0
n! 2! 3!
∞
X x2n+1 x3 x5
sin x = (−1)n =x− + − ···
n=0
(2n + 1)! 3! 5!
∞
X x2n x2 x4
cos x = (−1)n =1− + − ···
n=0
(2n)! 2! 4!
Shortcuts and Manipulations
• Substitution: Replace x with another expression. Example: ex = ∞ x2n
2 P
n=0 n!
.
(−1)n xn
• Negative Input: e−x = ∞
P
n=0 n!
.
• Integration: Integrate term by term. Example: ex dx = ∞ xn+1
R P
n=0 (n+1)! .
• Differentiation: Differentiate term by term. Example: d
dx
sin x = cos x matches the
series term by term.
P∞ n x4n+2
• Composition: Plug one series into another. Example: sin(x2 ) = n=0 (−1) (2n+1)! .
Applications
• Approximating values of functions near x = 0.
• Solving limits and indeterminate forms.
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