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 also 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 a sequence limit states that the limit of a sequence a n as n approaches infinity equals L if for every epsilon greater than zero, there exists an N such that for all n greater than N, the absolute value of a n minus L is less than epsilon. This means the terms get arbitrarily close to the limit. For the sequence 1 over n converging to zero, given any epsilon, we can find N equals 1 over epsilon, so that all terms beyond N are within epsilon of zero. Smaller epsilon values require larger N values.
Let's work through detailed convergence proofs. For the sequence 1 over n converging to 0, we need the absolute value of 1 over n minus 0 to be less than epsilon. This simplifies to 1 over n less than epsilon, which means n greater than 1 over epsilon. So N equals the ceiling of 1 over epsilon. For the sequence 2n plus 1 over n plus 3 converging to 2, we manipulate the expression to get the absolute value of negative 5 over n plus 3 less than epsilon. This gives us N equals the ceiling of 5 over epsilon minus 3. The graphs show how terms enter the epsilon neighborhoods around their respective limits.
Not all sequences converge. Divergent sequences fail to approach any finite limit. The sequence n grows without bound, making it impossible to find any finite L that satisfies the epsilon condition. The sequence negative 1 to the power n oscillates between negative 1 and positive 1, never settling to a single value. The sequence sine of n exhibits chaotic behavior with no discernible pattern. For a sequence to diverge, there must be no finite limit L such that for every epsilon greater than zero, we can find an N satisfying the convergence condition.
Limit theorems provide powerful tools for calculating sequence limits efficiently. 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 sequence b n is squeezed between sequences a n and c n, and both outer sequences converge to the same limit L, then b n also converges to L. For example, since negative 1 over n plus 1 and 1 over n plus 1 both approach 1, the squeezed sequence sine n over n plus 1 also approaches 1. These theorems simplify complex limit calculations significantly.