Polynomial lookalikes
A Maclaurin series is a power series—an infinite polynomial with carefully chosen coefficients—used to approximate functions that are differentiable near zero. The Taylor series generalizes this to approximations around any point where the function is finite and differentiable. The core idea is simple: match the function’s derivatives at a single point, and the polynomial will behave like the function near that point.
Suppose we want to approximate a function f(x). We start with a constant polynomial equal to f(0), so the approximation is exact at zero but poor elsewhere. Next, we set the first derivative of the approximation equal to f'(0), improving accuracy very close to zero in the direction of the slope. Continuing, we match the second derivative f''(0), then the third, and so on. If many derivatives of the polynomial agree with those of f at a point, the two functions will track each other closely around that point.
The general form of the Maclaurin series is:
As an example, computing the Maclaurin series for cos(x) is straightforward:
The dark blue line is cos(x); the lighter lines show successive approximations as more terms are added. With one term, the approximation is just the constant 1, the value of cos(0). With two terms it becomes a parabola. Each additional term tracks the true function more closely near zero, while farther from zero more terms are required for acceptable accuracy.
Deriving the coefficients
The Maclaurin series is built around zero for a practical reason: for small x, powers of x shrink quickly, helping the series converge. The coefficients are found by differentiating the power series term by term and evaluating at zero.
If we treat the function itself as the 0-th derivative, evaluating the power series and its derivatives at x=0 leaves only the constant term each time. Setting those equal to f(0), f'(0), f''(0), and so on determines the coefficients. Generalizing the pattern of denominators with factorials gives the definition of the Maclaurin series.
Why Taylor, not Maclaurin
Approximating around zero is not always the right choice. Two reasons motivate the shift:
- For a non-periodic function, we may want an accurate local approximation around a specific value of
x; a Maclaurin series’ quality degrades quickly with distance from zero. - The function may not be well-behaved near zero.
“Not well-behaved” typically means the function or its derivatives blow up at or near zero. The natural logarithm is the classic example: ln(x) is undefined at 0 and approaches minus infinity; every derivative 1/x, -1/x², 2/x³, and so on is also undefined at 0. The Maclaurin series simply cannot be formed. The Taylor series, however, works around any point a where the function is finite and differentiable.
For ln(x), approximating around a=1 works. Here ln(1)=0, and the derivatives at 1 follow a pattern: the n-th derivative is (-1)^{n+1}(n-1)!, giving an alternating sign. Substituting into the Taylor series formula yields a usable expansion:
The resulting plots show good agreement near x=1, but every finite approximation eventually diverges dramatically. The cause is explained by series convergence.
Convergence and the ratio test
To know whether a power series actually converges to the target function, and under what conditions, we can apply the ratio test. For a series, we compute the limit of the ratio of successive terms and require that it be less than 1 for absolute convergence.
For the Maclaurin series of cos(x), the ratio limit is independent of x and clearly satisfies the condition for all x. The series converges to cos(x) everywhere, which fits the fact that cosine is smooth on the whole real line.
For the Taylor series of ln(x) around x=1, the ratio test leads to an inequality whose solution is 0 < x < 2. Within this range the series converges; outside it, it does not. Equivalently, the radius of convergence around x=1 is 1, which explains the divergence seen in the plots beyond that interval.
