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Math Help - Testing Convergence

  1. #1
    Senior Member polymerase's Avatar
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    Testing Convergence

    Does:

    \displaystyle\sum_{n=1}^{\infty}\frac{\tan\left(\f  rac{1}{n}\right)}{n} converge?
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  2. #2
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    Quote Originally Posted by polymerase View Post
    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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