16.2: Convergence of Sequences

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Sequences

Definition: Sequence

A sequence is a function \(g_n\) defined on the positive integers '\(n\)'. We often denote a sequence by \(\left.\left\{g_{n}\right\}\right|_{n=1} ^{\infty}\)

Example \(\PageIndex{1}\)

A real number sequence:

\[g_{n}=\frac{1}{n} \nonumber \]

Example \(\PageIndex{2}\)

A vector sequence:

\[g_{n}=\left(\begin{array}{c}
\sin \left(\frac{n \pi}{2}\right) \\
\cos \left(\frac{n \pi}{2}\right)
\end{array}\right) \nonumber \]

Example \(\PageIndex{3}\)

A function sequence:

\[g_{n}(t)=\left\{\begin{array}{ll}
1 & \text { if } 0 \leq t<\frac{1}{n} \\
0 & \text { otherwise }
\end{array}\right. \nonumber \]

Note

A function can be thought of as an infinite dimensional vector where for each value of '\(t\)' we have one dimension

Convergence of Real Sequences

Definition: Limit

A sequnce \(\left.\left\{g_{n}\right\}\right|_{n=1} ^{\infty}\) converges to a limit \(g \in \mathbb{R}\) if for every \(\varepsilon > 0\) there is an integer \(N\) such that

\[\left|g_{i}-g\right|<\varepsilon, \quad i \geq N \nonumber \]

We usually denote a limit by writing

\[\operatorname{limit}_{i \rightarrow \infty} g_{i}=g \nonumber \]

\[g_{i} \rightarrow g \nonumber \]

The above definition means that no matter how small we make \(\varepsilon\), except for a finite number of \(g_i\)'s, all points of the sequence are within distance \(\varepsilon\) of \(g\).

Example \(\PageIndex{4}\)

We are given the following convergent sequence:

\[g_n=\frac{1}{n} \nonumber \]

Intuitively we can assume the following limit:

\[\operatorname{limit}_{n \rightarrow \infty} g_{n}=0 \nonumber \]

Let us prove this rigorously. Say that we are given a real number \(\varepsilon > 0\). Let us choose \(N=\left\lceil\frac{1}{\varepsilon}\right\rceil\), where \([x]\) denotes the smallest integer larger than \(x\). Then for \(n≥N\) we have

\[\left|g_{n}-0\right|=\frac{1}{n} \leq \frac{1}{N}<\varepsilon \nonumber \]

Thus,

\[\operatorname{limit}_{n \rightarrow \infty} g_{n}=0 \nonumber \]

Example \(\PageIndex{5}\)

Now let us look at the following non-convergent sequence

\[g_{n}=\left\{\begin{array}{l}1 \text { if } n=\text { even } \\ -1 \text { if } n=\text { odd }\end{array}\right. \nonumber \]

This sequence oscillates between 1 and -1, so it will therefore never converge.