Sequences of Functions

# Sequences of Functions

We have already looked at sequences of real numbers and sequences of elements in a metric space. We will now look specifically at sequences of functions, namely, real-valued functions.

 Definition: An Infinite Sequence of Functions $(f_n(x))_{n=1}^{\infty} = (f_1(x), f_2(x), ..., f_n(x), ...)$ is a sequence of functions with a common domain. The $n^{\mathrm{th}}$ Term of the sequence is the function $f_n(x)$.

We can define a finite sequence of functions analogously. A finite sequence of functions is denoted $(f_n)_{n=1}^{\infty}$.

We can also denote an infinite sequence of functions as simply $(f_n(x))$. We can also use curly brackets to denote a sequence of functions such as $\{ f_n(x) \}_{n=1}^{\infty}$ or simply $\{ f_n(x) \}$.

For example, consider the following sequence of functions:

(1)
\begin{align} \quad ( f_n(x) )_{n=1}^{\infty} = \left ( nx \right )_{n=1}^{\infty} = (x, 2x, ..., nx, ...) \end{align}

This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:

For another example, consider the following sequence of functions:

(2)
\begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^n)_{n=1}^{\infty} = (x, x^2, x^3, ..., x^n, ...) \end{align}

This is a sequence of the simplest $n^{\mathrm{th}}$ degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence: