Summary of Methods for Determining the Convergence of a Sequence

Summary of Methods for Determining the Convergence of a Sequence

We will now summarize some of the methods for determining whether a sequence is convergent. Let $A = (a_n)$ be a sequence of real numbers such that $\lim_{n \to \infty} a_n = L$.

  • Often times we can apply the definition of a sequence directly given a proposed limit, that is, we can show that $\forall \epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid < \epsilon$.
Theorem 1: Let $(a_n)$ be a convergent sequence. Then the following statements are equivalent:
1. The sequence $(a_n)$ converges to the real number $L$.
2. For every $\epsilon > 0$ there exists a natural number $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid < \epsilon$.
3. For every $\epsilon > 0$ there exists a natural number $N \in \mathbb{N}$ such that if $n ≥ N$ the terms satisfy $L - \epsilon < a_n < L + \epsilon$.
4. For every $\epsilon > 0$, for the $\epsilon$-neighbourhood $V_{\epsilon}(L)$ there exists a natural number $N \in \mathbb{N}$ such that if $n ≥ N$ then $x_n \in V_{\epsilon} (L)$.
  • If we notice that $(a_n)$ is a sequence that is obtained by square roots, absolutely values, or other properties of convergent sequences then we can invoke various convergent sequences theorems. For example, if we wanted to evaluate whether the sequence $\left ( \sqrt{ \frac{1}{n}} \right)$ converges or not, we could notice that $\left ( \frac{1}{n} \right )$ converges and since $\frac{1}{n} > 0$ for all $n \in \mathbb{N}$ it follows that $\left ( \sqrt{ \frac{1}{n}} \right)$ converges to $\sqrt{0} = 0$.
  • If we know that sequence lies in between two other sequences that converge to the same limit, then by the Squeeze theorem it follows that $(a_n)$ must also converge to that limit.
  • If $(a_n)$ is a sequence of positive real numbers and $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$ is easy to evaluate and $0 ≤ L < 1$, then we can invoke the Ratio test.
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