The Taylor Series: Some Common Taylor Series

1 + x +

+ ^... =

since e⁰ = 1. For any given x, the remainder term satisfies the inequality

| r_n(x)| =

e^c

= e^c

for some c between 0 and x. Since c≤x implies e^c≤e^x, a constant, and = 0, we once again have that | r_n(x)| = 0, so the function defined by the Taylor series for f (x) = e^x is equal to f (x) for all x.

The Logarithm Function

This time, since log(0) is not defined, we choose instead to find the Taylor series for f (x) = log(x) at x = 1. We compute the first few derivatives below:

f'(x)		=
f⁽²⁾(x)		= -
f⁽³⁾(x)		=
f⁽²⁾(x)		= -

Thus

f⁽ⁿ⁾(x) = (- 1)^n-1

for n≥1, so

f⁽ⁿ⁾(1) = (- 1)^n-1(n - 1)!

and the Taylor series for f (x) = log(x) at x = 1 is

(x - 1) -

+ ^... =

(- 1)^n-1

In this case, it turns out that the series only converges (to the value log(x)) for x in the interval (0, 2), i.e. the radius of convergence is 1. Letting y = x - 1, we may write