Testing Convergence

**polymerase** · Mar 30th 2008, 03:45 PM

Does:

$\displaystyle\sum_{n=1}^{\infty}\frac{\tan\left(\f rac{1}{n}\right)}{n}$ converge?

**ThePerfectHacker** · Mar 30th 2008, 05:17 PM

Originally Posted by polymerase

Does:

$\displaystyle\sum_{n=1}^{\infty}\frac{\tan\left(\f rac{1}{n}\right)}{n}$ converge?

Yes, it converges.

Note that,

$\frac{\tan \frac{1}{n}}{n} = \frac{\sin \frac{1}{n}}{n\cos \frac{1}{n}}$

Furthermore,

$\cos \frac{1}{n} \geq \kappa$ for all $n\geq 1$ , where $\kappa > 0$

Thus,

$\frac{\sin \frac{1}{n}}{n\cos \frac{1}{n}} \leq \frac{\sin \frac{1}{n}}{\kappa \cdot n} \leq \frac{\frac{1}{n}}{\kappa \cdot n} = \frac{1}{\kappa \cdot n^2}$

And the above series converges, now apply direct comparison.

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