Convergence of sin(1/n) Summation from n=1 to Infinity

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SUMMARY

The convergence of the series ∑ sin(1/n) from n=1 to infinity is established through the limit comparison test with the series ∑ 1/n. The limit comparison test shows that as n approaches infinity, sin(1/n) behaves similarly to 1/n, leading to the conclusion that the series converges. The correct application of the limit comparison test confirms that sin(1/n) converges, contrary to the initial assumption of divergence.

PREREQUISITES
  • Understanding of series convergence tests, specifically the limit comparison test.
  • Familiarity with the behavior of trigonometric functions as their arguments approach zero.
  • Knowledge of basic calculus, particularly limits and their properties.
  • Ability to manipulate and analyze infinite series.
NEXT STEPS
  • Study the limit comparison test in detail, including its conditions and applications.
  • Explore the behavior of sin(x) as x approaches zero to understand its implications in series.
  • Review examples of convergent and divergent series to solidify understanding of convergence criteria.
  • Investigate other convergence tests such as the ratio test and the root test for broader applications.
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Students studying calculus, particularly those focusing on series and convergence, as well as educators seeking to clarify convergence tests in mathematical analysis.

Frillth
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Okay, this is my last problem today. I swear.

Homework Statement



For this problem, I need to find the convergence of the sum of sin(1/n) from n=1 to infinity.

Homework Equations



None

The Attempt at a Solution



I know that this has to converge, but I'm having a hard time proving it. It seems like I should either be using the limit comparison test or just the comparison test, but I can't think of whta I can compare it to.
 
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This is very counter-intuitive. I just tried the limit comparison test with 1/n as follows:

lim sin(1/n) = cos(1/n) * -1/n^2 = cos(1/n) = 1
n->inf 1/n = -1/n^2

That means that sin(1/n) must diverge. Is this right?
 
I don't follow anything from your second post, except the conclusion, which is correct. But please describe more clearly how you arrived at it.
 

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