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Limits, Taylor and asymptotic series.

The program SERIES computes Taylor series, asymptotic development and limit at finite and infinite points. It should cover a lot of weird limits, even some that are not handled by the TI92 (not surprising!) nor by maple (more surprising!) like:

\begin{displaymath}\lim_{x\rightarrow 0} sin(1/x+x)-sin(1/x)\end{displaymath}

The LIMIT instruction may be used if you need only the limit. Note that SERIES handles more limits than LIMIT, but is a lot slower for trivial cases. SERIES can not be used with non-exact arguments (like 0.1) and should not be used with parameters. LIMIT may give strange results with parameters. If you see a warning message, you will get a binary integer as answer, this means that SERIES or LIMIT was not successful, the binary integer is an error code.

Syntax of SERIES:
Put on the stack the following arguments in this order

Type SERIES, this computes the bidirectional limit at level 3. At level 2, you get a list of two elements: the series expansion and the rest order. They are expressed in terms of a small parameter h. At level 1, h is expressed in terms of the initial variable (hence calling EXEC would return the series expansion and the rest in terms of the initial variable).

Remark 2  

Examples:
1/x $x=\infty $ SERIES
1/x $x=+\infty $ SERIES
1/x $x=-\infty $SERIES
$\sin(x)/x$ x SERIES
$\sin(x)/x$ $x=+\infty $SERIES
$\sqrt(2+x)$ x 5 SERIES
$\sin(1/x+x)-\sin(x)$ x SERIES
$( \ln(-\ln(x+x^2))-\ln(-\ln(x)) )*\ln(x)/x$ x=0+0 SERIES

The syntax of LIMIT is similar: put the function and the 'variable=limit_point' equation on the stack. Note that you can not force the order for series expansions and LIMIT handles only bidirectional limits (except at infinity). LIMIT returns only the limit at stack level 1.

Erable can handle relatively complex limits, like the example above (extracted from the Mupad on-line help):

\begin{displaymath}\frac{ \exp\left( \frac{e^{-x}}{e^{-x}+e^{\frac{-2x^2}{x+1} }} x \right)
- e^x } {x} \end{displaymath}

$X=\infty $ LIMIT returns -e2.

In addition Erable provides the:

These instructions may be used to understand series expansions without cumbersome calculations.


next up previous contents
Next: Derivation and integration. Up: Erable 3.024 Previous: Recurse flag.
Bernard Parisse
1998-07-31