# Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 556: 40

converges

#### Work Step by Step

Given $$\sum_{n=2}^{\infty} \frac{1}{n^2-\sqrt{n}}$$ Compare with the convergent series $\displaystyle \sum_{n=2}^{\infty}\frac{1}{n^2}$ and by using the Limit Comparison Test, we get: \begin{align*} \lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{n^2}{n^2-\sqrt{n}}\\ &=1 \end{align*} Then $\displaystyle \sum_{n=2}^{\infty}\frac{1}{n^2-\sqrt{n}}$ also converges

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