Taylor Series Mastery Worksheet | Calculus

📚 Taylor Series Formula:

f(x) = ∑ₙ₌₀^∞ [f⁽ⁿ⁾(a)/n!] · (x - a)ⁿ

Maclaurin Series: Special case with a = 0

Common Maclaurin Series:

eˣ = 1 + x + x²/2! + x³/3! + ...

sin(x) = x - x³/3! + x⁵/5! - ...

cos(x) = 1 - x²/2! + x⁴/4! - ...

Remainder Term (Lagrange):

Rₙ(x) = f⁽ⁿ⁺¹⁾(c)/(n+1)! · (x-a)ⁿ⁺¹

where c is between a and x

Problem 1: Maclaurin Series (Exponential) Beginner

f(x) = eˣ

Find the Maclaurin series expansion up to the 4th degree term. Use it to approximate e⁰·².

1 Derivatives at x = 0:

f(x) = eˣ ⇒ f(0) = 1

f'(x) = eˣ ⇒ f'(0) = 1

f''(x) = eˣ ⇒ f''(0) = 1

f'''(x) = eˣ ⇒ f'''(0) = 1

f⁽⁴⁾(x) = eˣ ⇒ f⁽⁴⁾(0) = 1

2 Maclaurin series formula:

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + f⁽⁴⁾(0)x⁴/4! + ...

3 Substitute values:

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

eˣ = 1 + x + x²/2 + x³/6 + x⁴/24 + ...

4 Approximate e⁰·² (x = 0.2):

e⁰·² ≈ 1 + 0.2 + (0.2)²/2 + (0.2)³/6 + (0.2)⁴/24

= 1 + 0.2 + 0.02 + 0.001333... + 0.0000666...

= 1.2213999...

5 Compare to actual value:

Actual e⁰·² ≈ 1.221402758... (calculator)

Error ≈ 0.000002858 (very small!)

💡 Insight: Even with just 5 terms, we get accuracy to 5 decimal places for x=0.2!

Problem 2: Taylor Series (Centered at π/2) Intermediate

f(x) = sin(x)

Find the Taylor series expansion centered at a = π/2 up to the 3rd degree term.

1 Derivatives and values at a = π/2:

f(x) = sin(x) ⇒ f(π/2) = 1

f'(x) = cos(x) ⇒ f'(π/2) = 0

f''(x) = -sin(x) ⇒ f''(π/2) = -1

f'''(x) = -cos(x) ⇒ f'''(π/2) = 0

f⁽⁴⁾(x) = sin(x) ⇒ f⁽⁴⁾(π/2) = 1

2 Taylor series formula:

f(x) = f(π/2) + f'(π/2)(x-π/2) + f''(π/2)(x-π/2)²/2! + f'''(π/2)(x-π/2)³/3! + f⁽⁴⁾(π/2)(x-π/2)⁴/4! + ...

3 Substitute values:

sin(x) = 1 + 0·(x-π/2) + (-1)(x-π/2)²/2! + 0·(x-π/2)³/3! + 1·(x-π/2)⁴/4! + ...

4 Simplify up to 3rd degree:

sin(x) ≈ 1 - (x-π/2)²/2 + (x-π/2)⁴/24 - ...

5 Check at x = π/2 + 0.1:

Actual: sin(π/2 + 0.1) = cos(0.1) ≈ 0.995004165...

Approximation: 1 - (0.1)²/2 = 1 - 0.005 = 0.995

Error ≈ 0.000004165 (excellent approximation!)

💡 Insight: When centered at π/2, sin(x) looks like an upside-down parabola near that point!

Problem 3: Maclaurin Series via Known Series Intermediate

f(x) = e⁻ˣ·cos(x)

Find the first 3 non-zero terms of the Maclaurin series by multiplying known series.

1 Known series expansions:

e⁻ˣ = 1 - x + x²/2! - x³/3! + x⁴/4! - ...

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

2 Multiply the series (up to x⁴ terms):

(1 - x + x²/2 - x³/6 + x⁴/24) × (1 - x²/2 + x⁴/24)

3 Collect terms by degree:

Constant: 1 × 1 = 1

x: 1 × 0 + (-x) × 1 = -x

x²: 1 × (-x²/2) + (-x) × 0 + (x²/2) × 1 = -x²/2 + x²/2 = 0

x³: 1 × 0 + (-x) × (-x²/2) + (x²/2) × 0 + (-x³/6) × 1 = x³/2 - x³/6 = x³/3

x⁴: 1 × (x⁴/24) + (-x) × 0 + (x²/2) × (-x²/2) + (-x³/6) × 0 + (x⁴/24) × 1 = x⁴/24 - x⁴/4 + x⁴/24 = -5x⁴/12

4 Result:

e⁻ˣ·cos(x) = 1 - x + (0)x² + (1/3)x³ - (5/12)x⁴ + ...

5 Verify with derivatives:

f(0) = 1, f'(0) = -1, f''(0) = 0, f'''(0) = 2, f⁽⁴⁾(0) = -10

Coefficients: f'''(0)/3! = 2/6 = 1/3 ✓, f⁽⁴⁾(0)/4! = -10/24 = -5/12 ✓

💡 Strategy: Multiplying known series is often easier than computing derivatives!

Problem 4: Lagrange Remainder Estimate Advanced

f(x) = ln(1 + x)

Using the 3rd degree Maclaurin polynomial, approximate ln(1.1). Estimate the error using Lagrange remainder.

1 Maclaurin series for ln(1+x):

ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ... for |x| < 1

2 3rd degree approximation (n=3):

P₃(x) = x - x²/2 + x³/3

For x = 0.1: P₃(0.1) = 0.1 - 0.01/2 + 0.001/3

= 0.1 - 0.005 + 0.000333... = 0.095333...

3 Lagrange remainder formula:

R₃(x) = f⁽⁴⁾(c)/4! · x⁴, where 0 < c < 0.1

4 Find f⁽⁴⁾(x):

f(x) = ln(1+x)

f'(x) = 1/(1+x)

f''(x) = -1/(1+x)²

f'''(x) = 2/(1+x)³

f⁽⁴⁾(x) = -6/(1+x)⁴

5 Maximum error bound:

Since f⁽⁴⁾(x) is decreasing, maximum |f⁽⁴⁾(c)| occurs at c = 0:

|f⁽⁴⁾(0)| = | -6/(1+0)⁴ | = 6

|R₃(0.1)| ≤ (6/24) · (0.1)⁴ = (1/4) · 0.0001 = 0.000025

6 Actual accuracy:

Actual ln(1.1) ≈ 0.095310179...

Our approximation: 0.095333...

Actual error: 0.000022821 < 0.000025 ✓

💡 Application: Lagrange remainder tells us our approximation is accurate to at least 4 decimal places!

📊 Essential Series Expansions

Function	Maclaurin Series	Radius of Convergence
eˣ	∑ xⁿ/n!	∞
sin(x)	∑ (-1)ⁿx²ⁿ⁺¹/(2n+1)!	∞
cos(x)	∑ (-1)ⁿx²ⁿ/(2n)!	∞
ln(1+x)	∑ (-1)ⁿ⁺¹xⁿ/n	1
1/(1-x)	∑ xⁿ	1

💪 Additional Practice Problems

Problem 5: f(x) = cos(2x) at a = 0
Answer: 1 - 2x² + 2x⁴/3 - ...

Problem 6: f(x) = √(1+x) at a = 0 (up to x³)
Answer: 1 + x/2 - x²/8 + x³/16 - ...

Problem 7: f(x) = arctan(x) at a = 0
Answer: x - x³/3 + x⁵/5 - ...

Problem 8: f(x) = sinh(x) at a = 0
Answer: x + x³/3! + x⁵/5! + ...