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### Section 10.10 : Ratio Test

2. Determine if the following series converges or diverges.

$\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!}}{{5n + 1}}}$

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We’ll need to compute $$L$$.

\begin{align*}L & = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{a_{n + 1}}}}{{{a_n}}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {{a_{n + 1}}\frac{1}{{{a_n}}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{\left( {2\left( {n + 1} \right)} \right)!}}{{5\left( {n + 1} \right) + 1}}\frac{{5n + 1}}{{\left( {2n} \right)!}}} \right|\\ & = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{\left( {2n + 2} \right)!}}{{5n + 6}}\frac{{5n + 1}}{{\left( {2n} \right)!}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{\left( {2n + 2} \right)\left( {2n + 1} \right)\left( {2n} \right)!}}{{5n + 6}}\frac{{5n + 1}}{{\left( {2n} \right)!}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{\left( {2n + 2} \right)\left( {2n + 1} \right)\left( {5n + 1} \right)}}{{5n + 6}}} \right| = \infty \end{align*}

When computing $${a_{n + 1}}$$ be careful to pay attention to any coefficients of $$n$$ and powers of $$n$$. Failure to properly deal with these is one of the biggest mistakes that students make in this computation and mistakes at that level often lead to the wrong answer!

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Okay, we can see that $$L = \infty > 1$$ and so by the Ratio Test the series diverges.