Sigma [sin(1/x)] for x=1 to ∞: Converge or Diverge?

[Phoenix314]

I need to determine whether Sigma [sin(1/x)] for x=1 to x=infinity converges or diverges. I have a feeling that it diverges, but I don't know how to prove it.

Thanks

 

[Pyrrhus]

Use the limit comparison test, with the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$

 

[Phoenix314]

Thanks Cyclovenom, I took $$\frac{sin(1/x)}{1/x}$$ and did the limit as x approaches infinity with L Hopital's rule, but I got 0, so doesn't that make it inconclusive?

Thanks

 

[Pyrrhus]

Your result through L'Hospital is wrong, it will be 1, and sice 1 > 0, and this harmonic series diverges, then sin (1/x) diverges.

Alternatively you could consider $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$

 

[steven187]

hello there

just use the integral test

$$\sum_{n=1}^{\infty} \sin{\frac{1}{n}} \le \int_{1}^{\infty}\sin{\frac{1}{x}}dx \not< \infty$$

its not finite and so does not converge

steven