AP Calculus BC

Taylor Series vs Power Series

Q: Does a power series have to be centred at zero?

No. The centre is the c in (x-c)^n, and zero is only the most common choice. The interval of convergence is always centred at c, so moving the centre moves the whole interval.

Every Taylor series is a power series. A power series is any sum of coefficients times powers of x minus c, with the coefficients free to be anything, and the only question you can ask is where it converges. A Taylor series is the one whose coefficients come from the derivatives of a named function at the centre.

Power series

Use when: You are handed a sum with unspecified coefficients and no function attached, and the question asks for the radius or interval of convergence.

Taylor series

Use when: A function is named, and the question asks for its series, a general term, or a polynomial approximation at a given centre.

Side by side

	Power series	Taylor series
Form	$\sum_{n=0}^{\infty} a_n (x-c)^n$	$\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$
Where the coefficients come from	Anywhere; they are given	Derivatives of $f$ at the centre $c$
Needs a function	No	Yes
What you are asked for	Radius and interval of convergence	The series, a general term, or an approximation
Common trap	Forgetting to test the two endpoints separately	Dropping the $n!$ under the derivative

The containment is the whole comparison. A power series is a shape, $\sum a_n (x-c)^n$ , and nothing about that shape says where the numbers $a_n$ came from. A Taylor series is that same shape with the coefficients pinned down: $a_n = \frac{f^{(n)}(c)}{n!}$ for a function you were handed.

The wording of the question tells you which one you are in. With no function in sight, the only questions available are convergence questions, and the ratio test on $\left\lvert \frac{a_{n+1}(x-c)^{n+1}}{a_n (x-c)^n} \right\rvert$ gives the radius, with the endpoints checked one at a time afterwards. With a function named, you are building coefficients, usually by substituting into a series you already know rather than differentiating five times.

The two meet in the middle

A power series with a positive radius of convergence defines a function on its interval, and it turns out to be that function's Taylor series centred at $c$ . So the objects coincide. What differs is the direction of work: from a function to coefficients, or from coefficients to an interval.

Frequently asked questions

Is every power series a Taylor series?

Any power series with a positive radius of convergence is the Taylor series of the function it sums to. The distinction that matters on an exam is whether a function was named: with one you compute coefficients, without one you can only test convergence.

Does a power series have to be centred at zero?

No. The centre is the $c$ in $(x-c)^n$ , and zero is only the most common choice. The interval of convergence is always centred at $c$ , so moving the centre moves the whole interval.

In the CED: Unit 10: Infinite Sequences and Series (BC)