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Section 10.11 : Root Test
2. Determine if the following series converges or diverges.
\[\sum\limits_{n = 0}^\infty {\frac{{{n^{1 - 3n}}}}{{{4^{2n}}}}} \]Show All Steps Hide All Steps
Start SolutionWe’ll need to compute \(L\).
\[L = \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left| {{a_n}} \right|}} = \mathop {\lim }\limits_{n \to \infty } {\left| {\frac{{{n^{1 - 3n}}}}{{{4^{2n}}}}} \right|^{\frac{1}{n}}} = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{n^{\frac{1}{n} - 3}}}}{{{4^2}}}} \right| = \left| {\frac{{{n^{\frac{1}{n}}}\,\,{n^{ - 3}}}}{{{4^2}}}} \right| = \frac{{\left( 1 \right)\left( 0 \right)}}{{16}} = 0\] Show Step 2Okay, we can see that \(L = 0 < 1\) and so by the Root Test the series converges.