A sequence is an ordered list of numbers, where each number corresponds to a positive integer index. Let's examine three examples. The sequence 1 over n approaches zero as n increases. The sequence negative 1 to the power n over n oscillates but still approaches zero. The sequence 2 plus 1 over n approaches 2. These examples show how sequences can converge to specific limit values as the index grows larger.
The formal definition of sequence convergence uses epsilon and N. A sequence converges to limit L if for every epsilon greater than zero, there exists an N such that all terms beyond N are within epsilon of L. Let's visualize this with the sequence 1 over n converging to zero. The red band shows the epsilon neighborhood around zero. As we make epsilon smaller, we need a larger N to ensure all subsequent terms fall within the band.
Let's work through detailed convergence proofs. For the sequence 1 over n converging to zero, we need the absolute value of 1 over n minus zero to be less than epsilon. This simplifies to 1 over n less than epsilon, so n must be greater than 1 over epsilon. For the sequence 2n plus 1 over n plus 3 converging to 2, we manipulate the expression to get 5 over n plus 3 less than epsilon. The graphs show how both sequences approach their respective limits, with the epsilon bands indicating the convergence regions.
Not all sequences converge. Divergent sequences fail to approach a single limit value. The sequence n grows without bound, making it impossible to find any finite limit L. The sequence negative 1 to the power n oscillates between negative 1 and positive 1, never settling on a single value. The sequence sine of n exhibits chaotic behavior, with values densely distributed throughout the interval from negative 1 to positive 1. These examples demonstrate the various ways sequences can fail to converge.
Limit theorems provide powerful tools for calculating sequence limits. The limit of a sum equals the sum of limits, the limit of a product equals the product of limits, and the limit of a quotient equals the quotient of limits when the denominator limit is non-zero. The squeeze theorem is particularly useful: if a sequence is bounded between two sequences that converge to the same limit, then the middle sequence also converges to that limit. For example, the sequence 3n squared plus 2n over 2n squared plus 1 can be simplified by dividing numerator and denominator by n squared, giving us 3 plus 2 over n divided by 2 plus 1 over n squared, which approaches 3 over 2.