Test Series for Convergence or Divergence

[flakine]

Summation, n=1 -> infinity, a(n)= sin(1/n)/n^(1/2)

Can I use the Limit Comparison test and make b(n) 1/n^(1/2)? The book says to make b(n) 1/n(n)^(1/2), but I can't understand why. Using my way the series diverge, but converge if I use the book's method.

 

[galactus]

\(\displaystyle \L\\\sum_{n=1}^{\infty}\frac{sin(\frac{1}{n})}{\sqrt{n}}\)


This series does converge.

As a matter of fact, it converges to 2.432932908