Sequences, Series, and Convergence Tests

Infinite Sequences

An infinite sequence is an ordered list of numbers that continues indefinitely. We denote a sequence as {a_n} or a₁, a₂, a₃, ..., where each a_n represents a term in the sequence. The value of each term is determined by its position n.

A sequence can be defined:

Explicitly: by giving a formula for the nth term (e.g., a_n = 1/n)
Recursively: by specifying initial terms and a rule for finding subsequent terms (e.g., a₁ = 1, a_n+1 = 2a_n)

Convergence of Sequences

A sequence {a_n} converges to a limit L if the terms of the sequence approach L as n approaches infinity. We write:

lim_n→∞ a_n = L

If a sequence does not converge, we say it diverges. A sequence might diverge by oscillating between values or by growing without bound.

Infinite Series

An infinite series is the sum of all terms in an infinite sequence. If {a_n} is a sequence, then the corresponding series is:

∑_n=1^∞ a_n = a₁ + a₂ + a₃ + ...

We define the partial sum S_n as the sum of the first n terms:

S_n = ∑_i=1ⁿ a_i = a₁ + a₂ + ... + a_n

An infinite series converges if the sequence of partial sums {S_n} converges to a finite limit S. We write:

∑_n=1^∞ a_n = S

If the sequence of partial sums does not converge to a finite limit, the series diverges.

Convergence Tests

Determining whether a series converges or diverges is a fundamental question in calculus. Several tests have been developed to help us answer this question:

1. The Divergence Test

If lim_n→∞ a_n ≠ 0, then the series ∑a_n diverges.

Note: If lim_n→∞ a_n = 0, the series may still diverge.

2. The Integral Test

Sequences, Series, and Convergence Tests