Mastering AP Calculus BC: Infinite Sequences and Series

Sequence

An ordered list of numbers, often generated by a formula.

Limit of a Sequence

The value that the terms of a sequence get arbitrarily close to as n approaches infinity.

Diverges

When a sequence does not converge to a limit.

Infinite Series

The sum of the terms of a sequence, denoted as the sum of a_n.

Sequence of Partial Sums

The sum of the first k terms of a sequence, denoted as S_k.

Convergence of a Series

A series converges if its sequence of partial sums approaches a limit.

Geometric Series

A series where each term is the previous term multiplied by a common ratio r.

Geometric Series Convergence Rule

Converges if |r| < 1; diverges if |r| ≥ 1.

Sum Formula for Geometric Series

If convergent, the sum is S = first term / (1 - r).

nth Term Test for Divergence

If lim an ≠ 0, the series diverges; if lim an = 0, it's inconclusive.

Integral Test

Uses an integral to determine convergence where f(x) models a_n and is positive, continuous, and decreasing.

p-Series Test

Converges if p > 1; diverges if 0 < p ≤ 1.

Harmonic Series

The series ∑(1/n) which diverges.

Direct Comparison Test (DCT)

Compares two series; if larger converges, smaller converges too.

Limit Comparison Test (LCT)

If a limit L of an/bn is positive and finite, both series share the same convergence behavior.

Alternating Series

Series whose terms alternate in sign, typically containing (-1)^n.

Alternating Series Test (AST)

Converges if lim bn = 0 and bn is decreasing.

Alternating Series Error Bound

The error in approximating the sum is bounded by the absolute value of the first unused term.

Absolute Convergence

Occurs when ∑|a_n| converges.

Conditional Convergence

Occurs when ∑an converges but ∑|an| diverges.

Ratio Test

Used for series with factorials or exponentials; checks limits of the ratio a{n+1}/an.

Power Series

A series defined by f(x) = ∑a_n(x-c)^n.

Radius of Convergence

Half the length of the interval in which the power series converges.

Interval of Convergence

The range of x values for which the power series converges.

Taylor Series

A series that represents a function around a point c using derivatives.

Maclaurin Series

Taylor series centered at c=0.

Metric Series (Geometric) Maclaurin Series

1/(1-x) = 1 + x + x^2 + … for |x|<1.

Exponential Maclaurin Series

e^x = 1 + x + x^2/2! + … for all x.

Sine Maclaurin Series

sin x = x - x^3/3! + x^5/5! - … for all x.

Cosine Maclaurin Series

cos x = 1 - x^2/2! + x^4/4! - … for all x.

Lagrange Error Bound

Bounds the error of a Taylor polynomial approximation using the next derivative.

Common Mistake: Limit of Sequence vs. Series

Finding lim an = 0 does NOT guarantee ∑ an converges.

Common Mistake: Harmonic Series Trap

The Harmonic Series diverges despite lim a_n = 0.

Endpoint Testing in Power Series

Must test endpoints individually for convergence.

Condition for Alternating Series Test

Check that terms in AST are decreasing when considering magnitudes.

Lagrange M Confusion

Use the (n+1) derivative for error bounds in Taylor polynomial approximations.

Error in Partial Sums

If the series converges, the error of using a partial sum is less than the next term.

Example of a Converging Series

The series 3(1/2)^n converges with a sum of 6.

Example of Diverging Series

The Harmonic series diverges even though its terms go to zero.

p-Test Convergence

p-series converges if p is greater than 1.

Convergence of Exponential Series

The series for e^x converges for all x.

Estimating Convergence with Power Series

Check the radius using the ratio of coefficients for convergence.

Condition for series convergence

Both series must converge or diverge by the Integral Test.

Trigonometric Maclaurin Series

Sine and Cosine series differ in the powers of terms.

Behavior of Series Under Comparison

Inference about convergence can be made using direct and limit comparison tests.

Polynomial Approximation Error

Error in Taylor series can be determined by derivative bounds.

Usage of Series in Function Representation

Series like Taylor allow complex functions to be expressed as polynomials.