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".
[itex]\lim_{n=0} sin(2\pi n)= 0[/itex], trivially.
Bipolarity said:I think there is some confusion between [itex] sin(2\pi n) [/itex] 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.
The convergence of a sequence is a mathematical concept that describes the behavior of a sequence as its terms approach a specific limit or value. It means that as the sequence progresses, the terms get closer and closer to the specified limit.
While both involve the idea of approaching a limit, the main difference between the convergence of a sequence and the convergence of a series is that a sequence is a list of numbers, whereas a series is the sum of all the terms in a sequence. This means that the convergence of a series depends on the convergence of the individual terms in the sequence.
A convergent sequence is a sequence that has a limit or a finite value that the terms of the sequence approach. On the other hand, a divergent sequence is a sequence that does not approach a specific limit and instead has terms that either grow infinitely large or oscillate between different values without approaching a limit.
To determine the convergence of a sequence, you can use the definition of convergence, which states that a sequence is convergent if and only if its terms approach a specific limit. This can be checked by calculating the limit of the sequence using various methods, such as algebraic manipulation, substitution, or using theorems like the squeeze theorem or the ratio test.
The convergence of a sequence is a fundamental concept in mathematics as it allows us to study the behavior of infinite sets of numbers. It has applications in various areas of mathematics, including calculus, analysis, and topology, and is essential in understanding the behavior and properties of functions, series, and other mathematical objects.