A Taylor polynomial matches a function’s value and derivatives at a single point.
Step 1: Let's Learn
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Maclaurin series
A Taylor series centred at zero. Nothing else distinguishes it.
The ones worth knowing
eˣ = 1 + x + x²/2! + …, and sin x = x − x³/3! + x⁵/5! − …
Where it is valid
A series equals its function only inside the interval of convergence. Outside it, the two are unrelated.
A function rebuilt from its derivatives at one point
The Taylor series matches the function's value and every derivative at a single point. Each term corrects the previous approximation, and the polynomial degree can be raised as far as accuracy demands.
The formula
The coefficient of (x − a)ⁿ is f⁽ⁿ⁾(a)/n!. The factorial appears because differentiating a power n times produces n!, which must be divided out for the coefficients to match.
The series worth memorising
eˣ, sin x, cos x and 1/(1 − x). Most BC series questions are handled by manipulating one of these rather than computing derivatives from scratch, which is far slower.
The Lagrange error bound
The remainder after n terms is bounded by the next term with the derivative replaced by its maximum on the interval. Error questions expect this bound and expect the maximum to be justified.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Omar Build Three Terms
Omar writes the Maclaurin polynomial of eˣ to three terms.
- Step 1
Every derivative of eˣ is eˣ, which is 1 at x = 0.
Step 4: Your Turn
Practice makes it stick.
The Factorial
Problem 1 of 2
What is 4! ?
The Centre
Problem 2 of 2
A Maclaurin series is centred at which value?
Building the Series
1 of 8
What is 3! ?
2 of 8
What is 5! ?
3 of 8
In the series for eˣ, what is the coefficient of x²? Give the denominator.
4 of 8
The series for sin x has only which powers? 1 for odd, 2 for even.
5 of 8
The series for cos x has only which powers? 1 for odd, 2 for even.
6 of 8
A Taylor series centred at 3 is called a Maclaurin series. 1 for yes, 0 for no.
7 of 8
Evaluate 1 + x + x²/2 at x = 0.
8 of 8
Select every true statement about Taylor series.
Step 5: Quick Check
Show what you know.
Question 1 of 1
What is 6! ?
What You Learned
- A Taylor polynomial matches a function’s derivatives at one point.
- A Maclaurin series is simply a Taylor series centred at zero.
- The series represents the function only inside its interval of convergence.