Taylor & Maclaurin Series Calculator

Taylor Series Guide

What Is a Taylor Series?

A Taylor series expresses a function as an infinite sum of polynomial terms. For a function f(x) about point a: f(x) = f(a) + f'(a)(x−a)/1! + f''(a)(x−a)²/2! + f'''(a)(x−a)³/3! + ... When a=0, this is the Maclaurin series. Common series: eˣ = 1 + x + x²/2! + x³/3! + ... (all x). sin(x) = x − x³/3! + x⁵/5! − ... (all x, x in radians). cos(x) = 1 − x²/2! + x⁴/4! − ... (all x, x in radians). ln(1+x) = x − x²/2 + x³/3 − ... (−1 < x ≤ 1). 1/(1−x) = 1 + x + x² + x³ + ... (|x| < 1).

Why Taylor Series Matter?

Taylor series are one of the most powerful tools in mathematics, physics, and engineering. Approximation: for small x, sin(x) ≈ x (first term). This small-angle approximation is used throughout physics — pendulum motion, optics, relativistic corrections. Numerical computation: computers evaluate transcendental functions (sin, cos, exp, ln) using polynomial approximations derived from Taylor series — faster to compute than the exact function. Error estimation: the remainder term bounds how wrong the polynomial approximation can be, giving a mathematically rigorous guarantee of accuracy rather than just an intuitive estimate.

What should I know about Convergence and Radius?

A Taylor series only equals the original function within its radius of convergence. eˣ: converges for all x (radius = ∞). sin(x), cos(x): converges for all x. ln(1+x): converges for −1 < x ≤ 1. 1/(1−x): converges for |x| < 1. Outside the radius: the series diverges (partial sums grow without bound). At the boundary (x at the edge of radius): convergence may be conditional. Alternating series test: an alternating series (terms alternate in sign) converges if terms decrease monotonically to zero — this is the alternating series test.

What's the key thing to understand about Applications in Physics?

Small-angle approximation: sin(θ) ≈ θ for θ < 0.2 rad (~11°). Error < 1% for θ < 10°. Used in: simple harmonic motion equations, small-signal amplifier analysis, thin lens formula derivation. Binomial series: (1+x)ⁿ ≈ 1 + nx + n(n−1)x²/2 for |x| < 1. Used in: relativity (expanding the Lorentz factor), uncertainty analysis, AC circuit approximations. Euler's formula: eⁱˣ = cos(x) + i·sin(x). This emerges directly from comparing the Taylor series of eˣ, cos(x), and sin(x). The most beautiful equation in mathematics, according to many mathematicians, emerges as a special case: eⁱπ + 1 = 0, connecting five fundamental constants in a single elegant expression.

What should I know about the Maclaurin Series (Taylor Series at x=0)?

A Maclaurin series is simply a Taylor series expanded around a=0, which simplifies the general formula f(x) = Σ f^(n)(a)/n! × (x−a)ⁿ to f(x) = Σ f^(n)(0)/n! × xⁿ. The most useful standard Maclaurin series are worth memorising: eˣ = 1+x+x²/2!+x³/3!+..., sin(x) = x−x³/3!+x⁵/5!−..., cos(x) = 1−x²/2!+x⁴/4!−..., and ln(1+x) = x−x²/2+x³/3−... (valid for −1<x≤1). These series are how calculators and computers actually compute trig and exponential functions internally — a chip has no 'sine circuit', only arithmetic, so it sums enough terms of the series to reach the required precision. For small x, just the first two or three terms often give excellent accuracy: sin(0.1) ≈ 0.1 − 0.1³/6 = 0.09983, matching the true value to four decimal places.

What do I need to know about Common Taylor Series Mistakes?

The most frequent error is using too few terms far from the expansion point — a Taylor series is most accurate near a, and accuracy degrades (sometimes badly) as x moves further away, unless the series has an infinite radius of convergence (like eˣ, sin(x), cos(x), which converge everywhere). Series for functions like ln(1+x) or 1/(1−x) have a finite radius of convergence and simply don't represent the function outside that radius, no matter how many terms you add. Another common mistake is forgetting the factorial in the denominator of each term, or miscounting derivatives — the nth derivative of sin(x) cycles through sin, cos, −sin, −cos every four steps, and a mistake tracking which one you're on at a given n throws off every subsequent term. Always double-check the first two or three terms of a famous series (eˣ, sin x, cos x) against their known forms before trusting a derived expansion for a less common function.

What should I know about Common Maclaurin Series Worth Memorizing?

A handful of Maclaurin series (Taylor series centered at x = 0) come up so often that they're generally worth memorizing outright rather than re-deriving each time: eˣ = 1 + x + x²/2! + x³/3! + ..., sin(x) = x - x³/3! + x⁵/5! - ..., cos(x) = 1 - x²/2! + x⁴/4! - ..., and 1/(1-x) = 1 + x + x² + x³ + ... (valid for |x| < 1). Noticing the pattern between eˣ's series and sin(x)/cos(x)'s series — same terms, different signs and which powers are included — is also what underlies Euler's formula, e^(ix) = cos(x) + i·sin(x), one of the most celebrated results connecting exponential, trigonometric, and complex functions. Having these memorized makes it much faster to approximate a function's value near zero, or to recognize a series in a problem without computing derivatives from scratch.

What should I know about a Taylor Series Reference Table?

The most useful standard Maclaurin series, with their radius of convergence, gathered for quick lookup (note this includes two series not detailed elsewhere on this page):

What should I know about a Worked Example: Approximating cos(0.3)?

As a worked numerical example, approximate cos(0.3) using the first three terms of its Maclaurin series: cos(x) ≈ 1 − x²/2! + x⁴/4!. Substituting x=0.3: 1 − (0.09)/2 + (0.0081)/24 = 1 − 0.045 + 0.0003375 = 0.9553375. The true value of cos(0.3) (to 7 decimal places) is 0.9553365. The approximation's error is only about 0.000001 — roughly one part in a million — demonstrating how rapidly Taylor series converge for small x, since only two or three terms are needed for excellent precision close to the expansion point.

What should I know about Taylor Series Around a Nonzero Point?

All the Maclaurin series above are Taylor series expanded specifically at a=0, but the general Taylor series formula works around any point a: f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + .... Expanding around a nonzero point is useful when a function isn't even defined at zero, such as f(x)=ln(x), or when high accuracy is needed near a specific value far from zero. Worked example: find the Taylor series for ln(x) around a=1. The derivatives at a=1 are f(1)=0, f'(1)=1, f''(1)=−1, f'''(1)=2, giving ln(x) ≈ (x−1) − (x−1)²/2 + (x−1)³/3 − .... Using this to estimate ln(1.2) with three terms: (0.2) − (0.2)²/2 + (0.2)³/3 = 0.2−0.02+0.00267 = 0.18267, very close to the true value of ln(1.2) ≈ 0.18232.

What should I know about Big-O Notation and Truncation Error?

When a Taylor series is truncated after a finite number of terms to create a usable approximation, the size of the error left behind is often described using big-O notation. Writing sin(x) = x − x³/6 + O(x⁵) means that after keeping the first two terms, the error introduced is proportional to x⁵ (and smaller) as x shrinks toward zero — the next omitted term in the actual series. This notation compactly states how quickly an approximation's error shrinks as you move closer to the expansion point. For example, halving x shrinks an O(x⁵) error by a factor of 2⁵=32, while an O(x²) error would only shrink by a factor of 4 for the same halving — meaning a series whose error term has a higher power of x converges far more rapidly near the expansion point, which is exactly why extra terms in a Taylor series are most valuable close to the point of expansion and increasingly unhelpful far away. This is also why numerical software libraries that compute sin, cos, or exp internally switch to range-reduction tricks (shifting x into a small interval near zero first) rather than simply adding ever more terms to a series expansion centred far from the input value.