Unit 10BC

Infinite Series

BC Only17–18% of BC ExamCED Unit 10

Foundations

Sequences, series, partial sums, and geometric series: the building blocks of Unit 10.

Sequence

Definition: an ordered list of values where n is the index and a_n is the nth term

A sequence is a function whose domain is the positive integers.

Convergence of a Sequence

Limit of the nth Term

To find the limit of a sequence, find a formula for the

th term using

th term expansion, then take the limit as

If the limit of the

th term exists and is finite, the sequence converges to that number. Otherwise it diverges.

Series

A series is the sum of the terms of a sequence.

Finite Series

Infinite Series

Partial Sums

The

th partial sum of a series is the sum of the first

terms of

nth Partial Sum

The list

is called the sequence of partial sums.

Convergence of an Infinite Series

The Key Equation

If this limit is finite, the series converges to

, and

is the sum of the series.

If the limit does not exist or is infinite, the series diverges.

Convergence of a Geometric Series

Geometric Series

A geometric series has a constant ratio

between successive terms.

Sum Formula

Condition	Behavior	Sum
	Converges
	Diverges	–

Identify a (first term) and r (common ratio). Check

first.

Critical Fact

This is the starting point for power series manipulation (CED 10.14, 10.15).

⚠ Watch Out

and

are not the same

is an individual term.

is the running total of all terms up to

. These are different sequences.

⚠ Watch Out

does not guarantee convergence

The harmonic series

has

but diverges. This is the single most common misconception in the series unit.

AP® Calculus BC · Unit 10 Overview · Mr. Brantley