 RateOfConvergence - Maple Help

Student[NumericalAnalysis]

 RateOfConvergence
 compute the rate of convergence of a sequence or function Calling Sequence RateOfConvergence(S, n, opts) Parameters

 S - algebraic; the function or sequence n - name; the name of the indexing variable opts - (optional) equation(s) of the form keyword = value where keyword is isfunction; whether S is is a function or sequence Options

 • isfunction = true or false
 Whether to find the rate of convergence of a sequence or a function.  If S is a function, isfunction must be set to true. By default, isfunction=false. Description

 • If the input is a sequence ${a}^{n}$, the return value of the RateOfConvergence command is of the form $L+\mathrm{O}\left({n}^{-d}\right)$, where $L=\underset{n\to \mathrm{\infty }}{lim}{a}_{n}$ and ${a}_{n}-L=\mathrm{O}\left({n}^{-d}\right)$ for some $0.
 • If the input is a function $f\left(x\right)$, the return value is of the form $L+\mathrm{O}\left({x}^{d}\right)$, where $L=\underset{x\to 0}{lim}f\left(x\right)$ and $f\left(x\right)-L=\mathrm{O}\left({x}^{d}\right)$ for some $0
 • If S is a function, the option isfunction must be set to true.
 • By default, the isfunction option is set to false. Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right):$
 > $\mathrm{RateOfConvergence}\left(1+\frac{1}{n},n\right)$
 ${1}{+}{\mathrm{O}}{}\left(\frac{{1}}{{n}}\right)$ (1)
 > $\mathrm{RateOfConvergence}\left(\frac{1}{2}-\frac{1}{{n}^{2}},n\right)$
 $\frac{{1}}{{2}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{n}}^{{2}}}\right)$ (2)
 > $\mathrm{RateOfConvergence}\left(\mathrm{exp}\left(x\right),x,\mathrm{isfunction}=\mathrm{true}\right)$
 ${1}{+}{\mathrm{O}}{}\left({x}\right)$ (3)