# How do you use the ratio test to test the convergence of the series #∑ (n!)^2 / (kn)!# from n=1 to infinity?

The series:

is convergent for

In our case:

Now we have that:

and the series is divergent.

and the test is inconclusive, so we have to look at the series in more detail:

So we have:

And as:

is convergent, then also our series is convergent by direct comparison.

and the series is convergent.

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To use the ratio test to test the convergence of the series ∑ (n!)^2 / (kn)! from n=1 to infinity, you compute the limit as n approaches infinity of the absolute value of the ratio of consecutive terms in the series, which is given by:

lim (n→∞) |[a_{n+1} / a_n]|

For the given series, the nth term is ((n!)^2) / ((kn)!). Therefore, the (n+1)th term is (((n+1)!)^2) / ((k(n+1))!).

Substitute these into the ratio test formula and simplify to find the limit. If the limit is less than 1, the series converges. If it is greater than 1, the series diverges. 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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