Function: math[PSconv] - convergence of power series
Calling Sequence:
PSconv(s);
PSconv(s, option);
Parameters:
s -
a power series defined with function Sum s :=
option - (optional) either 'limit' or 'range'
Description:
In the first form, PSconv determines the radius r of convergence of a power series s,
r :=
In this case, the result is a realcons.
With option 'limit' PSconv returns
where
is simplified before.
With option 'range' PSconv determines the range of convergence. The range of convergence is established as follows: If r =
, then PSconv returns 'real', otherwise the function solves the relation
or
. If r=0 then the function returns a value of type
realcons
, if r > 0 it returns a
RealRange
expression (like
solve
returns with relations). In this latter case the function checks whether the power series s converges at the border elements.
NOTE
: Since function
sum
- which is used by PSconv to evaluate the border elements
left
,
right
of the range of convergence - returns incorrect results with
alternating geometric sums
, (
), PSconv returns an incorrect result with this option, e.g. instead of Open(
left
) it returns
left
(see example
below). If
sum
returns an unevaluated expression when determining convergence of the power series s at the border elements, a warning message is displayed. The border elements have to be checked with other means.
This function is part of the math package, and so can be used only after performing the command with(math) or with(math, PSconv).
Examples:
> with(math, PSconv):
> s := Sum((-1)^(n+1)*(x-1)^n/n, n=1 .. infinity); # -> 0 < x <= 2
> PSconv(s);
> PSconv(s, limit);
> PSconv(s, range);
> s := Sum((x-1)^(n-1)*n!, n=1 .. infinity); # -> r=0
> PSconv(s);
> PSconv(s, range);
Now a geometric series:
> s := Sum(x^n, n=1 .. infinity); # -> -1 < x < 1
> PSconv(s);
> PSconv(s, limit);
The correct result is RealRange(Open(-1), Open(1)), but PSconv returns:
> PSconv(s, range);
> sum((-1)^n, n=1 .. infinity);
> seq((-1)^n, n=1 .. 15);
See Also:
sum , share,analysis/FPS , convert/range .
Version History:
Version 1.0 as of November 02, 1997
Version 1.0.1 as of July 17, 2000