Limit of the sequence

Definition

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. It is denoted as $\lim_{{n \to \infty}} a_n = L$.

5 Must Know Facts For Your Next Test

1. A sequence $\{a_n\}$ has a limit $L$ if for every $\epsilon > 0$, there exists an integer $N$ such that for all $n \geq N$, $|a_n - L| < \epsilon$.
2. If a sequence converges to a limit, it must be bounded.
3. A sequence can have at most one limit.
4. Not all sequences have limits; those that do not are called divergent.
5. The Squeeze Theorem can be used to find limits of certain sequences.

Review Questions

• What is the definition of the limit of a sequence?
• How would you prove that a sequence converges to a specific limit?
• Can a divergent sequence be bounded? Explain.

Related terms

Convergence: A sequence is said to converge if it approaches some finite limit as the index goes to infinity.

Divergence: A sequence is said to diverge if it does not approach any finite limit as the index goes to infinity.

Squeeze Theorem: If $b_n \leq a_n \leq c_n$ and both $\lim_{{n \to \infty}} b_n = \lim_{{n \to \infty}} c_n = L$, then $\lim_{{n \to \infty}} a_n = L$.