Sequences and Infinite Series

Sequences and their Limits

A sequence is an ordered collection of elements from a set \(S\). We can denote a sequence with \(a_i\) or \(\{a_i\}\) where \(i\) is a natural number, i.e., \(i\in\mathbb{N}\). We can also list the first few elements, e.g., \(a_1, a_2, a_3, \ldots\) The elements of a sequence does not have to be single numbers. They can be vectors, in which case we have a sequence of vectors. Formally, a sequence is a function \(f\) that takes an element from the set of natural numbers and associates with it an element of another set \(S\), i.e., \(f:\mathbb{N}\rightarrow S\).

Convergence of Sequences

We say that a sequence \(\{a_i\}\) converges if it approaches some limit \(a\). Formally,

\[\lim_{i\rightarrow\infty}a_i=a\]

if, for any \(\epsilon>0\), there exists \(N\), such that if \(i>N\), then \(\lvert a_i-a\rvert<\epsilon\). To get an intuition about this statement, we visualize a convergent sequence in the plot below.

Notice how, after \(N\) terms, subsequent terms are confined inside the upper and lower bounds (gray lines). A sequence is convergent if this happens for any \(\epsilon\), no matter how small it is.

If the sequence does not converge, we say that it is divergent.

Properties

• If two sequences, \(a_i\) and \(b_i\), converge to \(a\) and \(b\), respectively, then:

\[a_i+b_i\rightarrow a+b\]

\[a_ib_i\rightarrow ab\]

• If \(g\) is a continuous function, then

\[g(a_i)\rightarrow g(a).\]

Infinite Series

Given an infinite sequence \(\{a_i\}\), we add the first \(n\) terms to get a series. An infinite series is then the limit of this sum as \(n\) goes to infinity, i.e.,

\[\lim_{n\rightarrow\infty}\sum_{i=1}^{n}a_i=\sum_{i=1}^{\infty}a_i\]

provided this limit exists.

Geometric Series

The geometric series is given by

\[S=\sum_{i=0}^{\infty}\alpha^i=1+\alpha+\alpha^2+\ldots\]

This series will converge if \(\lvert\alpha\rvert<1\), in which case we have

\[S=\dfrac{1}{1-\alpha}.\]