How do you apply the ratio test to determine if #Sigma (4^n(n!)^2)/((2n)!)# from #n=[1,oo)# is convergent to divergent?
The ratio test is indeterminate for this series.
For the series:
Evaluate the ratio:
So we have:
which means the ratio test is indecisive and we cannot determine whether the series is convergent or not.
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To apply the ratio test to determine the convergence or divergence of the series ( \sum_{n=1}^{\infty} \frac{{4^n(n!)^2}}{{(2n)!}} ):

Compute the ratio of consecutive terms: [ \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right ] where ( a_n = \frac{{4^n(n!)^2}}{{(2n)!}} ).

If the limit is less than 1, the series converges. If it's greater than 1 or does not exist, the series diverges.

If the limit equals 1, the test is inconclusive, and other tests may be necessary.

Compute the limit: [ \lim_{n \to \infty} \left \frac{{a_{n+1}}}{{a_n}} \right = \lim_{n \to \infty} \left \frac{{4^{n+1}((n+1)!)^2}}{{(2(n+1))!}} \cdot \frac{{(2n)!}}{{4^n(n!)^2}} \right ]
[ = \lim_{n \to \infty} \left \frac{{4(n+1)^2}}{{(2n+2)(2n+1)}} \right ]
[ = \lim_{n \to \infty} \left \frac{{4(n^2 + 2n + 1)}}{{4n^2 + 6n + 2}} \right ]
[ = \lim_{n \to \infty} \left \frac{{n^2 + 2n + 1}}{{n^2 + \frac{3}{2}n + 1}} \right ]
[ = 1 ]
Since the limit equals 1, the ratio test is inconclusive. Additional tests, such as the root test or comparison test, may be needed to determine the convergence or divergence of the series.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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