Taylor & Maclaurin Series – Differential Calculus for GATE, ESE, AE/JE

Taylor and Maclaurin Series are important topics in Differential Calculus used to approximate functions. They are frequently asked in GATE, ESE, AE/JE exams and are very scoring when standard expansions are known.

Basic Concept

A function can be expanded as an infinite series using derivatives.

Taylor Series

Expansion of f(x) about x = a:

f(x) = f(a) + (x−a)f’(a) + (x−a)²/2! f’’(a) + (x−a)³/3! f’’’(a) + ...

Maclaurin Series

Special case of Taylor series when a = 0:

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

Standard Expansions (VERY IMPORTANT)

• eˣ = 1 + x + x²/2! + x³/3! + ...
• sin x = x − x³/3! + x⁵/5! − ...
• cos x = 1 − x²/2! + x⁴/4! − ...
• ln(1+x) = x − x²/2 + x³/3 − ...
• 1/(1−x) = 1 + x + x² + x³ + ...

Applications

• Finding limits
• Approximating functions
• Solving indeterminate forms
• Used in higher calculus problems

Solved Examples

Example 1

Expand eˣ up to x³

1 + x + x²/2 + x³/6

Example 2

Expand sin x up to x⁵

x − x³/6 + x⁵/120

Example 3

Find:

lim (x → 0) (eˣ − 1 − x) / x²

eˣ = 1 + x + x²/2 + ...

Numerator = x²/2

Answer: 1/2

Example 4

lim (x → 0) (sin x − x) / x³

sin x = x − x³/6 + ...

Numerator = −x³/6

Answer: −1/6

Shortcut Tricks

• Memorize standard expansions
• Use series only up to required power
• Ignore higher order terms
• Very useful in limit problems

Questions

Q1. Expansion of eˣ starts with?

Answer: 1 + x

Q2. Expansion of ln(1+x)?

Answer: x − x²/2 + x³/3 − ...

Exam Tips

• Focus on standard series
• Apply in limit questions
• Avoid unnecessary full expansion
• Practice PYQs

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