Mathfolis

Power Series

Unit 10 · Infinite Sequences & Series

Power Series and Taylor/Maclaurin Series

A power series ∑cₙ(x−a)ⁿ converges on an interval centered at a with radius R found via the Ratio Test. Always check both endpoints separately. Taylor polynomials approximate f(x) near a using f and its derivatives; the Lagrange error bound |Rₙ(x)| ≤ M|x−a|ⁿ⁺¹/(n+1)! controls the approximation error.

Taylor/Maclaurin Polynomial

Pn(x)=∑k=0nf(k)(a)k!(x−a)kP_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k

Standard Maclaurin Series

ex=∑n=0∞xnn!e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}
sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}
cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}
11−x=∑n=0∞xn,∣x∣<1\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n,\quad |x|<1

Lagrange Error Bound

∣Rn(x)∣≤M⋅∣x−a∣n+1(n+1)!|R_n(x)| \le \frac{M\cdot|x-a|^{n+1}}{(n+1)!}
M = max|f⁽ⁿ⁺¹⁾(c)| for c between a and x
Caution: After finding the radius R with the Ratio Test, always check BOTH endpoints x = a ± R individually — convergence there is not determined by the Ratio Test (L = 1).
AP Tip: Derive new Maclaurin series by substitution into a known series, or by differentiating/integrating term by term. After differentiating/integrating, re-check endpoint convergence.
Type 1

Taylor/Maclaurin Polynomial Construction

Compute successive derivatives at a, apply the formula, then use the polynomial to approximate the function.

Example 1

Find the degree-4 Maclaurin polynomial for cos x and use it to approximate cos(0.2).

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Type 2

Lagrange Error Bound

Identify the (n+1)th derivative, find its maximum M on the interval between a and x, then apply the error formula.

Example 2

Bound the error when approximating eˣ with P₃(x) = 1 + x + x²/2 + x³/6 at x = 0.5.

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Type 3

Radius and Interval of Convergence

Apply the Ratio Test to find R. The series converges on (a−R, a+R). Then test each endpoint individually.

Example 3

Find the radius and interval of convergence for ∑xⁿ/(n·2ⁿ).

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Type 4

Representing Functions as Power Series

Start from 1/(1−x) = ∑xⁿ and derive new series by substitution, differentiation, or integration.

Example 4

Write f(x) = 1/(1+3x) as a power series centered at 0.

Example 5

Express ln(1+x) as a power series.

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