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### Section 10.11 : Root Test

1. Determine if the following series converges or diverges.

$\sum\limits_{n = 1}^\infty {{{\left( {\frac{{3n + 1}}{{4 - 2n}}} \right)}^{2n}}}$

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We’ll need to compute $$L$$.

$L = \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left| {{a_n}} \right|}} = \mathop {\lim }\limits_{n \to \infty } {\left| {{{\left( {\frac{{3n + 1}}{{4 - 2n}}} \right)}^{2n}}} \right|^{\frac{1}{n}}} = \mathop {\lim }\limits_{n \to \infty } \left| {{{\left( {\frac{{3n + 1}}{{4 - 2n}}} \right)}^2}} \right| = {\left( { - \frac{3}{2}} \right)^2} = \frac{9}{4}$ Show Step 2

Okay, we can see that $$L = \frac{9}{4} > 1$$ and so by the Root Test the series diverges.