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## Limits for sequences: Example

Maple

With a member of the community I had some discussion about using Maple for limits of sequences.

The specific task was about F:=4*sqrt(n)*sin(Pi*sqrt(4*n^2+sqrt(n))) and limit F for n --> infinity for integers, which is asserted to be Pi, see https://math.stackexchange.com/questions/2493385/find-the-limits-lim-n4-sqrtn-sin-pi-sqrt4n2-sqrtn.

For moderate size of integers n it can be 'confirmed' by numerical evaluations. Formally it is not 'obvious' at all.

limit(F, n = infinity) assuming n::posint;

undefined

This is 'explained' by the help about assuming/details:

"The assuming command does not place assumptions on integration or summation dummy variables in definite integrals and sums, nor in limit or product dummy variables, because all these variables already have their domain restricted by the integration, summation or product range or by the method used to compute a limit. ..."
The help continues with suggestions how to treat the situation.

Which means that the limit is taken in the Reals, not in the discrete Naturals.

A more simple example may be limit(sin(n*Pi), n = infinity) assuming n::posint which returns "-1 .. 1".

But here we go:

MultiSeries:-asympt(F, n):
simplify(%) assuming n::posint: collect(%, n); #lprint(%);
limit(%, n=infinity);

O(1/2048*Pi^3/n^(5/2)) + Pi -1/96*Pi^3/n - 1/16*Pi/n^(3/2) + 1/30720*Pi^5/n^2

Pi

As desired.

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