The discussion revolves around the convergence or divergence of the series given by the sum of (n sin n) / (n^3 + 1) from n equals 1 to infinity. Participants are exploring the behavior of this series as n approaches infinity.
The discussion is active, with participants questioning each other's reasoning and clarifying the distinction between limits and sums. Some guidance has been offered regarding the simplification of terms and the implications of the oscillatory behavior of sin n.
There is a noted confusion between evaluating the limit of the sequence and the sum of the series, which has led to some miscommunication among participants.
-EquinoX- said:Homework Statement
the sum of (n sin n)/(n^3+1)
n from 1 to infinity
Homework Equations
The Attempt at a Solution
what I have done so far is using the comparison method by comparing this with n sin n/ n^3, but I don't know what's the next step and even if my comparison is correct or not
-EquinoX- said:that's where I was stuck. I think it doesn't go anywhere therefore it doesn't exist. Am I right??
-EquinoX- said:well because sin n varies between -1 and 1. This what makes me think that the limit does not converges to some value
-EquinoX- said:it goes to infinity, so therefore it goes to 0?? Am I right?
-EquinoX- said:yeah If I replace it with either -1 or 1 both will most likely to reach 0 from both sides, from the negative sides and the positive sides.
-EquinoX- said:well we can simplify [n * sin n]/n^3 into sin n/n^2 right?? therefore the answer is 0?
-EquinoX- said:what do you mean by deduce?
HallsofIvy said:Excuse me, but the original question asked for the SUM of [itex]\frac{nsin(n)}{n^3+1}[/itex], not the limit of the sequence.