The Manga Guide to Calculus

154 Chapter 5 Let’s Learn About Taylor Expansions!

When a general function f(x) (provided it is differentiable infinitely many

times) can be expressed as

f x a a x a x a x a x

n

n

( )

= + + + + + +

0 1 2

2

3

3

... ...

the right side is called the Taylor expansion of f(x) (about x = 0).

Left side =

−

= −

1

1 2

1

Right side

= + + + + +

1 2 4 8 16 ...

It turns out that expression u is correct for all

x satisfying −1 < x < 1, which is the allowed interval

of a Taylor expansion. In technical terms, the inter-

val −1 < x < 1 is called the circle of convergence.

S? The two

sides are not

equal.

For example, substituting x = 2

in both sides of expreion u,

This means tha