Divergence Criteria

The Divergence Criteria for Sequences

Thus far we have looked at criteria for sequences to be convergent. We will now begin to look at some criteria which will tell us if a sequence is divergent.

Theorem 1 (Divergence Criteria for Sequences): A sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} of real numbers is divergent if either one of the following hold:
1. $(a_{n})$ has two subsequences $(a_{n_{k}})$ and $(a_{n_{p}})$ that converge to two different limits.
2. $(a_{n})$ has a subsequence that is divergent.

3. $(a_{n})$ is unbounded.

Notice that if either (1) or (2) hold then this immediately contradicts the fact that if a sequence $(a_{n})$ is convergent then all of its subsequences $(a_{n_{k}})$ converge to the same limit.

Recall by <a href="/the-boundedness-of-convergent-sequences-theorem">The Boundedness of Convergent Sequences Theorem</a> that if a sequence is convergent that it is bounded. The contrapositive of this statement is that is a sequence is not bounded then it is divergent, and so then (3) is justified as well.

We will now look at some examples of apply the Divergence Criteria for Sequences.

Example 1

Show that the sequence $((-1)^{n})$ is divergent.

To show this sequence is divergent, consider the subsequence of even terms which is $(1,1,1,...)$ which converges to the real number $1$. Now consider the subsequence of odd terms which is $(-1,-1,-1,...)$ which converges to the real number $-1$.

Since these two subsequences converge to different limits, we conclude that $((-1)^{n})$ is divergent.

Example 2

Show that the sequence $(a_{n})$ defined by $a_{n}=\left\{{\begin{matrix}1/n&{\text{if n is even}}\\n&{\text{if n is odd}}\end{matrix}}\right.$.

Let's first look at a few terms of this sequence. We have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n) = \left(1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, ... \right)} . We can see this sequence is not bounded above and hence not bounded, which we will prove.

Suppose that there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \in \mathbb{N}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a_n \mid M} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} . By the Archimedean property since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \in \mathbb{R}} there exists a natural number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_M \in \mathbb{N}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \leq n_M} . We also note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \leq n_M n_M + 1} . So either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_M} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_M + 1} is a term in the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} which contradicts the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} from being bounded.

Example 3

Show that the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)} is divergent.

Once again, this sequence is unbounded. Suppose that instead Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)} is bounded, that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid n \mid = n M} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \in \mathbb{R}} . But this contradicts the Archimedean property which says that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \in \mathbb{R}} there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_M \in \mathbb{N}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \leq n_M} , and so in fact Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)} is not bounded and by the divergence criteria, is divergent as well.

Resources

• The Divergence Criteria for Sequences, mathonline.wikidot.com