What is the coefficient formula for a Maclaurin series centered at 0?

Conditions

• The series is expanded around $x=0$.
• The function must have derivatives up to order n at 0 for the displayed formula to make sense.
• The function is assumed to have a power series representation.

Reasoning, step by step

1. Start with a power series $f(x) = c_0 + c_1x + c_2x^2$ + ... + c_nx^n + ...
2. Differentiate the series term by term n times to get $f^{(n)}(x) = n$! * $c_n$ + ...
3. Substitute $x=0$ into the nth derivative formula. All terms containing x vanish, leaving $f^{(n)}(0) = n$! * $c_n$.
4. Solve for $c_n$ by dividing both sides by n!, yielding $c_n = f^{(n)}(0) / n$!.

Example

The board writes $c_n = f^{(n)}(0) / n$! after substituting $x=0$ into the derivative hierarchy. The speaker says generically $c_n$ is obtained by taking n derivatives and dividing by n! to balance the expression.

Common misconceptions

• Forgetting to divide by n!, which would incorrectly set $c_n$ equal to the nth derivative itself.
• Believing that the formula applies to series centered at an arbitrary point a without replacing 0 with a in the derivative evaluation.

Watch the explanation

Explore next

• Why does the nth derivative of a power series produce a factor of n! in front of $c_n$?
• What is the difference between a Maclaurin series and a Taylor series?