Chapter 8: Infinite Sequences and Series

Section 8.3: Convergence Tests

Example 8.3.4

$$ Determine if the series $\sum _{n\=1}^{\infty }\frac{n}{\sqrt{n^2\+2}}$ diverges, converges absolutely, or converges conditionally. If it converges conditionally, determine if it also converge absolutely. $$ Solution $$ For large $n$, $n\/\sqrt{n^2\+2}$ behaves like $n\/n\=1$, suggesting that the limiting behavior of $a_n$ be tested by   $\underset{n\to \infty }{lim}\frac{n}{\sqrt{n^2\+2}}$ = $1$    By the nth-term test, since $a_n$ does not go to zero as $n\to \infty$, the series must necessarily diverge. $$

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