Meet Taylor Series

Today we are meeting Taylor Series.

Taylor Series are power series for functions with

derivatives of all orders. They are important for

extending the domains of functions to include

complex numbers.

Let f be a function with derivatives of all orders in

some interval containing a as the interior point. The

Taylor series generated by f at a is

Sum from k=0 to infinity of (f(k’) (a) (x — a)^k / k!)

= f(a) + f’(a) (x — a) + f’’(a) (x — a)² / 2!

• f’’’(a) (x — a)³ / 3! + … (1)

The Maclaurin series is the Taylor series at x = 0

f(0) + f’(0) x + f’’(0) x² / 2! + f’’’(0) x³ / 3! + … (2)

Taylor’s Theorem:

If f and its first n derivatives f’, f”, f’’’, …, f(n’) are

continuous on [a , b] and f(n’) is differentiable on

(a , b) , then there exists c between a and b such that

f(b) = f(a) + f’(a) (b — a) + f’’(a) (b — a)² / 2! + …

• f(n’) (a) (b — a)^n / n! + f(n+1') ( c)(b — a)^(n+1) / (n+1)! (3)

Example 1: f(x) = e^x

f’ = (d/dx) e^x = e^x and f” = e^x and f’’’ = e^x …

Let a = 0 , f’ = f” = f’’’ = e⁰ = 1

From eqn(2)

e^x = 1 + x + x² /2! + x³ /3! + …

Example 2: f(x) = sin x

f’ = cos x , f” = — sin x , f’’’ = — cos x

Let a = 0 , f’ = 1 , f” = 0 , f”’ = — 1 , f(0) = 0

From eqn(2)

sin x = x — x³ /3! + x⁵ /5! — x⁷ /7! + …

Example 3: f(x) = cos x

f’ = — sin x , f” = — cos x , f’’’ = sin x

Let a = 0 , f’ = 0 , f” = — 1 , f”’ = 0 , f(0) = 1

From eqn(2)

cos x = 1 — x² /2! + x⁴ /4! — x⁶ /6! + …

Example 4: From Ex 1

e^x = 1 + x + x² /2! + x³ /3! + …

find e^(iw)

e^(iw) = 1 + iw + (iw)² /2! + (iw)³ /3! + (iw)⁴ /4!

• (iw)⁵ /5! + (iw)⁶ /6! + …

= (1 — w² /2! + w⁴ /4! — w⁶ /6! + …)

• i(w — w³ /3! + w⁵ /5! — …)

So e^(iw) = cos w + i sin w (4)

Eqn(4) is Euler’s formula

When w = pi , e^(i pi) = — 1

So e^(i pi) + 1 = 0 (5)

Amazing! Equation (5) combines 5 most important

constants in Mathematics.

Please enjoy and have fun! Thank you!