Convergence & Divergence

Unit 10 · Infinite Sequences & Series

Convergence and Divergence of Series

An infinite series ∑aₙ converges if its sequence of partial sums Sₙ approaches a finite limit. The nth Term Test says: if lim aₙ ≠ 0, the series diverges. If lim aₙ = 0, the test is inconclusive — the harmonic series ∑1/n is the classic counterexample (terms → 0, yet the series diverges).

Partial Sums and Convergence

Partial sum

S_n = a_1 + a_2 + \cdots + a_n

Series converges if

\lim_{n\to\infty} S_n = S \text{ (finite)}

nth Term Test for Divergence

\text{If } \lim_{n\to\infty} a_n \ne 0, \text{ then } \sum a_n \text{ diverges.}

Caution: The nth Term Test can only prove divergence. If lim aₙ = 0, you cannot conclude convergence — use a different test.

AP Tip: For telescoping series, use partial fractions to write aₙ as a difference, then watch intermediate terms cancel in Sₙ.

Type 1

nth Term Test for Divergence

Compute the limit of the general term. If the limit is nonzero, the series diverges.

Example 1

Determine whether ∑(2n² + 1)/(n² + 5) converges or diverges.

Type 2

Telescoping Series

Decompose the general term using partial fractions so adjacent terms cancel in the partial sum, leaving a simple limit.

Example 2

Find the sum of ∑[n=1 to ∞] 1/((n+1)(n+2)).

Type 3

Convergence from a Partial Sum Formula

When given a formula for Sₙ directly, find the series sum by taking the limit as n → ∞.

Example 3

The partial sums of a series are Sₙ = 3n/(n+1). Find S₁, S₂, S₃ and determine the sum of the series.

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