Function: math[PSconv] - convergence of power series

Calling Sequence:

PSconv(s);

PSconv(s, option);

Parameters:
s - a power series defined with function Sum s := [Maple Math]

option - (optional) either 'limit' or 'range'

Description:

In the first form, PSconv determines the radius r of convergence of a power series s,

r := [Maple Math]

In this case, the result is a realcons.

With option 'limit' PSconv returns

[Maple Math]

where [Maple Math] is simplified before.

With option 'range' PSconv determines the range of convergence. The range of convergence is established as follows: If r = [Maple Math] , then PSconv returns 'real', otherwise the function solves the relation [Maple Math] or [Maple Math] . 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 , ( [Maple Math] ), 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

[Maple Math]

> PSconv(s);

[Maple Math]

> PSconv(s, limit);

[Maple Math]

> PSconv(s, range);

[Maple Math]

> s := Sum((x-1)^(n-1)*n!, n=1 .. infinity); # -> r=0

[Maple Math]

> PSconv(s);

[Maple Math]

> PSconv(s, range);

[Maple Math]

Now a geometric series:

> s := Sum(x^n, n=1 .. infinity); # -> -1 < x < 1

[Maple Math]

> PSconv(s);

[Maple Math]

> PSconv(s, limit);

[Maple Math]

The correct result is RealRange(Open(-1), Open(1)), but PSconv returns:

> PSconv(s, range);

[Maple Math]

> sum((-1)^n, n=1 .. infinity);

[Maple Math]

> seq((-1)^n, n=1 .. 15);

[Maple Math]

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