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Common Maclaurin Series

Definition

Common Maclaurin series are the standard power series expansions (centered at x=0x=0) for fundamental transcendental functions. They serve as the “alphabet” for representing more complex mathematical expressions as infinite polynomials.

Why It Matters

They provide a universal ‘alphabet’ for approximating transcendental functions, making complex calculus problems solvable by calculators and computers.

Core Concepts

  • Exponential function (the prototype) ex=∑n=0∞xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

    • How to read: “The function e to the x equals the sum from n equals zero to infinity of x to the n all over n factorial.”
    • Meaning / convergence / use: The coefficients 1/n! come from f^{(n)}(0) = 1 for all derivatives of e^x. Converges for every real (and complex) x — the series is its own definition in many rigorous treatments. Use for computing e^0.1 accurately by hand, for solving differential equations, or for the Poisson distribution.

  • Sine (odd function) sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!=x−x33!+x55!−x77!+⋯\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

    • How to read: “The sine of x equals the sum from n equals zero to infinity of the quantity negative one to the n times x to the quantity two n plus one, all over the quantity two n plus one factorial.”
    • Meaning: Only odd powers appear because sine is an odd function: sin(−x) = −sin x. The alternating sign comes from the cycle of derivatives (sin → cos → −sin → −cos → sin …). Truncating after a few terms gives excellent small-angle approximations (sin x ≈ x − x³/6).

  • Cosine (even function) cos⁡x=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!−x66!+⋯\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

    • How to read: “The cosine of x equals the sum from n equals zero to infinity of the quantity negative one to the n times x to the two n, all over the quantity two n factorial.”
    • Meaning: Only even powers — cosine is even. Same derivative cycle produces the alternating signs. cos x ≈ 1 − x²/2 is the starting point for the small-angle pendulum approximation and for the Fresnel integrals.

  • Geometric series (finite and infinite) 11−x=∑n=0∞xn(∣x∣<1)\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad (|x| < 1)

    • How to read: “The expression one divided by the quantity one minus x equals the sum of x to the n from zero to infinity, where the absolute value of x is less than one.”
    • Meaning / use: The most important series in mathematics. Every term is the previous multiplied by x. It is the closed form for the infinite geometric sum. Used constantly in generating functions, in the formula for the sum of an infinite geometric series, in Taylor expansions of 1/(1−x), and as the kernel of the geometric distribution.

  • Natural logarithm ln⁡(1+x)=∑n=1∞(−1)n−1xnn(−1<x≤1)\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} \quad (-1 < x \leq 1)

    • How to read: “The natural log of the quantity one plus x equals the sum from n equals one to infinity of the quantity negative one to the n minus one, times x to the n, all over n.”
    • Meaning: This is the Taylor series for ln at 1. It converges (slowly) at x=1 to ln 2. The alternating version is useful for |x| < 1. Classic way to compute logarithms or to integrate 1/(1+x) term by term.

  • Arctangent arctan⁡x=∑n=0∞(−1)nx2n+12n+1(∣x∣≤1)\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \quad (|x| \leq 1)

    • How to read: “The arctangent of x equals the sum from n equals zero to infinity of the quantity negative one to the n times x to the quantity two n plus one, all over the quantity two n plus one.”
    • Meaning / famous special case: At x=1 this becomes the Leibniz formula π/4 = 1 − 1/3 + 1/5 − 1/7 + …. Extremely slow convergence but historically important. The series is used in many computer implementations for arctan near zero and in some Fourier series derivations.

Practical pattern to remember: Look at the powers (even vs odd), the signs (alternating for sin/cos/arctan), and the factorial or plain n in the denominator. These five series plus the binomial series cover the vast majority of hand approximations and limit evaluations in calculus.

Connected Concepts

Connected notes