# What is meant by “x” in the nth term of a sequence

By definition, a sequence is a function whose domain is positive integers. When we are given the $n^{th}$ term of a sequence, we calculate $a_1$ when $n=1$, $a_2$ when $n=2$ and so on, we can graph these numbers on $XY$ plane by considering $Y$ axis as $a_n$ and $X$ axis as $n$, but when we have $x$ in the $n^{th}$ term of the sequence (as in $a_n=(\frac{x^n}{2n+1})^{1/n})$ what does it mean? shall we consider it ("$x$") as a real function? then in that case the sequence becomes a composite function($f\circ g$). Now, the real function "may" have entire real line as its domain and its range, and the range of the inside function becomes the domain of the outside function in composite function and it does not agree with the definition of a sequence, so what went wrong? If I want to plot $a_n=(\frac{x^n}{2n+1})^{1/n})$ for $x>0$what values $x$ and $n$ will take?

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Can't you/one think of $x$ like a parameter? –  draks ... Apr 2 '12 at 11:36

In this particular case, I don't think a plot is possible. For example

$a_1 = x/3$

$a_2 = (x^2/5)^{1/2}$

$a_3 = (x^3/7)^{1/3}$

This is a sequence, not of numbers, but of functions.

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Your expression $a_n=\left(\frac{x^n}{2n+1}\right)^{1/n}$ corresponds to a sequence of functions rather than a composition of functions. That is, as a sequence it is a function whose domain is $\mathbb{N}$ and whose range is the space of all functions $\mathbb{R}\to \mathbb{R}.$ For $n=1$ you get the function $x\mapsto \frac{x}{3};$ for $n=2$ the function $x\mapsto \left(\frac{x^2}{5}\right)^{1/2},$ and so on. These sequences of functions (and the corresponding series of functions, e.g. power series) show up very often; in elementary calculus they are usually studied in relation with problems of pointwise vs. uniform convergence. Plotting the sequence usually means plotting each one of the functions (well, really the first few of them) on the same XY plane, to get an idea of what these functions tend to look like when $n\to \infty.$

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Do you have some context for this sequence?

In this case you might take the sequence $a$ to be a function of both $x$ and $n$. So a has the domain of (Reals $\times$ Naturals) and range of (Reals)

You would then have three dimensions (2 dimensional input, 1 dimensional output), and would need to plot this on a 3-D plot.

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or, as per @Juan S's answer, you could plot each term on a regular 2-D graph, having one graph for each term. –  Ronald Apr 2 '12 at 11:40
thank you friends, I am relived! –  Vikram Apr 2 '12 at 11:42