# How do you apply the ratio test to determine if #Sigma (3^n(n!)^2)/((2n)!)# from #n=[1,oo)# is convergent to divergent?

Prove that:

The ratio test states that a necessary condition for the series:

to converge is that

Calculate the expression of the ratio for this series:

Passing to the limit:

so the series is convergent.

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To apply the ratio test to determine the convergence or divergence of the series ( \sum_{n=1}^\infty \frac{3^n(n!)^2}{(2n)!} ), calculate the limit as ( n ) approaches infinity of the absolute value of the ratio of the ( (n+1) )th term to the ( n )th term. If the limit is less than 1, the series is convergent; if it's greater than 1, the series is divergent; if it equals 1, the test is inconclusive.

Using the ratio test:

[ \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{{n \to \infty}} \left| \frac{\frac{3^{n+1}((n+1)!)^2}{(2(n+1))!}}{\frac{3^n(n!)^2}{(2n)!}} \right| ]

After simplification:

[ \lim_{{n \to \infty}} \left| \frac{3^{n+1}(n+1)^2(n+1)!^2(2n)!}{(2(n+1))!3^nn!^22n!} \right| ]

This simplifies to:

[ \lim_{{n \to \infty}} \left| \frac{3(n+1)^2(n+1)!^2}{(2n+2)(2n+1)(2n+2)(2n+1)} \right| ]

This further simplifies to:

[ \lim_{{n \to \infty}} \left| \frac{3(n+1)(n+1)!^2}{(2n+2)^2(2n+1)^2} \right| ]

Now, as ( n ) approaches infinity, the limit of this expression can be evaluated. If the limit is less than 1, the series converges; if it's greater than 1, the series diverges.

After calculation, if the limit is less than 1, then the series ( \sum_{n=1}^\infty \frac{3^n(n!)^2}{(2n)!} ) is convergent. If it's greater than 1, the series is divergent. If it equals 1, the test is inconclusive.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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