10.1 Defining Convergent and Divergent Infinite Series

Infinite series and partial sums

An infinite series is the sum of the terms of an infinite sequence: a₁+a₂+a₃+.... To study its behavior, we build partial sums Sₙ = a₁+a₂+...+aₙ, which form a new sequence. The infinite series' sum is defined as the limit of this sequence of partial sums, not just simple addition forever.

Convergence vs. divergence

A series converges if the sequence of partial sums Sₙ approaches a finite limit S as n→∞; then we say the series' sum equals S. If the limit of Sₙ doesn't exist or is infinite, the series diverges. This is the foundational definition used before applying any convergence tests.

Using partial sums to test behavior

To determine convergence, write an explicit formula for Sₙ if possible, then compute limₙ→∞ Sₙ. Geometric series and telescoping series are classic examples where Sₙ can be found explicitly. If Sₙ oscillates, grows unboundedly, or has no limit, the series diverges.

Key terms

Infinite series

The sum a₁+a₂+a₃+... of the terms of an infinite sequence.

Partial sum (Sₙ)

The sum of the first n terms of a series: Sₙ = a₁+a₂+...+aₙ.

Convergent series

A series whose sequence of partial sums approaches a finite limit as n→∞.

Divergent series

A series whose sequence of partial sums does not approach a finite limit.

Sum of an infinite series

Defined as limₙ→∞ Sₙ, the limit of the partial sums, when that limit exists.

Try an AP-style question

A series has partial sums Sₙ = 1 − 1/(n + 1). What is the sum of the series?

1. 1
2. 0
3. 1/2
4. ∞