Common Maclaurin Series
Series for eˣ, sin x, cos x, ln(1+x).
Key idea
Series for eˣ, sin x, cos x, ln(1+x).
In depth
Taylor series centred at 0 are worth memorising: eˣ = Σxⁿ/n!, sin x = Σ(−1)ⁿx²ⁿ⁺¹/(2n + 1)!, cos x = Σ(−1)ⁿx²ⁿ/(2n)!, 1/(1 − x) = Σxⁿ, ln(1 + x) = Σ(−1)ⁿ⁺¹xⁿ/n.
Worked example
sin x ≈ x − x³/6 for small x. 1/(1 − x) ≈ 1 + x + x².
Formula
Real-world uses
- Quickly approximating e^x or sin x in small-angle physics calculations
Common mistake
Using sin series for cos or mixing odd and even powers.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Common Maclaurin Series?
Series for eˣ, sin x, cos x, ln(1+x). It is a topic at the expert level in Calculus.
What is the formula for Common Maclaurin Series?
The key formula is eˣ = Σ xⁿ/n!.
Where is Common Maclaurin Series used in real life?
Real-world examples: Quickly approximating e^x or sin x in small-angle physics calculations.
What should I learn before Common Maclaurin Series?
Make sure you are comfortable with Taylor Series, the previous topic in Calculus.
What is a common mistake with Common Maclaurin Series?
Using sin series for cos or mixing odd and even powers.
How can I practise Common Maclaurin Series?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Common Maclaurin Series?
Move on to Taylor’s Remainder next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.