Taylor and Maclaurin Series

f

has a power series representation

\sum c_n(x-a)^n

near

a

, then repeatedly differentiating and evaluating at

x = a

forces

c_n = f^{(n)}(a)/n!

. This is the Taylor series of

f

a

. The key message: the Taylor series is uniquely determined by the derivatives of

f

at one point, and for "nice" functions (analytic functions) the series equals

f

on its entire interval of convergence. The Maclaurin series of

e^x

\sin x

\cos x

, and

(1+x)^k

are the four workhorses that appear throughout finance:

e^x

in the Black-Scholes exponential;

\sin x

and

\cos x

in Fourier pricing (characteristic function methods);

(1+x)^k

in binomial approximations.

Learning objectives. By the end of this note you should be able to:

derive the Taylor/Maclaurin series formula from the coefficient equation;
write out the standard Maclaurin series for $e^x$ , $\sin x$ , $\cos x$ , and $(1+x)^k$ ;
multiply and compose power series to find new ones;
identify when $T_n(x)$ is a useful approximation to $f(x)$ .

In this lesson

The full lesson, with worked examples and exercise solutions, is for members.