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 sequence convergence uses epsilon and N. A sequence converges to limit L if for every epsilon greater than zero, we can find an N such that all terms beyond N are within epsilon of L. For the sequence 1 over n converging to zero, choosing smaller epsilon requires larger N values. Watch how the epsilon neighborhood shrinks and the N threshold moves right.
Let's work through three convergence examples with detailed proofs. First, the sequence 1 over n converges to zero. We need the absolute value of 1 over n minus zero to be less than epsilon, which gives us n greater than 1 over epsilon. Second, the sequence 2n plus 1 over n plus 3 converges to 2. After algebraic manipulation, we get n greater than 5 over epsilon minus 3. Third, sine of n over n converges to zero using the squeeze theorem, since the absolute value is bounded by 1 over n.