Power Series: The Essentials

// Your cheat sheet for the most important formulas and concepts. No fluff.

Core Concepts

General Form: A power series is centered at a.
∑ c_n(x - a)ⁿ
Radius of Convergence (R): The distance from the center that the series converges.
Interval of Convergence (I.O.C.): The full range of x values where the series converges.
Most Important Rule: You must ALWAYS TEST THE ENDPOINTS of the interval.

The Ratio Test: Your Main Tool

Use the Ratio Test to find the radius. The series converges if L < 1.

L = lim_n→∞ |

a_n+1 a_n

|

Three Possible Outcomes:

If L = 0, the series converges for all x. Then R = ∞.
If L = ∞, the series converges only at its center, x = a. Then R = 0.
If L = K |x - a|, then the radius is R = 1/K.

Taylor & Maclaurin Series

The formula for representing a function as a power series.

Taylor Series centered at a:
f(x) = ∑
f⁽ⁿ⁾(a) n!
(x - a)ⁿ
Maclaurin Series is a Taylor Series centered at a = 0.

Essential Maclaurin Series to Memorize

Function	Series Expansion	Interval
11 - x	1 + x + x² + x³ + ...	(-1, 1)
e^x	1 + x + x²/2! + x³/3! + ...	(-∞, ∞)
sin(x)	x - x³/3! + x⁵/5! - ...	(-∞, ∞)
cos(x)	1 - x²/2! + x⁴/4! - ...	(-∞, ∞)
ln(1 + x)	x - x²/2 + x³/3 - ...	(-1, 1]
arctan(x)	x - x³/3 + x⁵/5 - ...	[-1, 1]

Key Rules of Thumb ("Tricks")

The center (a) comes from the (x - a)ⁿ term.
The radius (R) comes from the other exponential parts (like kⁿ).
Polynomial parts like n^k do not affect the radius.
If you see n! in the denominator, the radius is almost always R = ∞.
If you see n! in the numerator, the radius is almost always R = 0.