The discussion revolves around the convergence or divergence of the sequence defined by a_{n} = sin(2πn). Participants are exploring the behavior of this sequence as n approaches infinity.
There is an ongoing exploration of the implications of limits in different contexts, with some participants providing insights into the relationship between limits for natural and real numbers. Confusion regarding the convergence of the sequence is acknowledged, but no consensus has been reached.
Participants note the potential confusion arising from the definitions of the sine function for different domains and the implications for determining convergence.
LCKurtz said:Write out the numerical value of a few terms of that sequence.
HallsofIvy said:I have no idea what you mean by "when we apply the limit it diverges".
\lim_{n=0} sin(2\pi n)= 0, trivially.
Bipolarity said:I think there is some confusion between sin(2\pi n) defined for natural numbers, and that defined for real numbers. If we take the limit of it as it is defined for natural numbers, it converges to 0 trivially. But if we take the limit of it as it is defined for all real x, its limit does not exist.
Does this mean we cannot take the limit of the function defined for reals to determine the convergence of the function (or sequence) defined for naturals?
BiP
LCKurtz said:Yes and no. If the limit exists over the reals then it exists over the naturals. The converse is false. Also the limit may exist over the naturals but not over the reals as that example shows.