# How do you test for convergence for #1/((2n+1)!) #?

In the case you meant "test the convergence of the *series* :

the Answer is : it

In our case

Next we test,

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To test for convergence of the series ( \frac{1}{(2n+1)!} ), you can use the Ratio Test. The Ratio Test states that for a series ( \sum_{n=1}^{\infty} a_n ), if ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 ), then the series converges absolutely.

For the given series, [ a_n = \frac{1}{(2n+1)!} ]

So, [ \frac{a_{n+1}}{a_n} = \frac{\frac{1}{(2(n+1)+1)!}}{\frac{1}{(2n+1)!}} = \frac{(2n+1)!}{(2(n+1)+1)!} = \frac{1}{(2n+3)(2n+2)(2n+1)!} ]

Taking the limit as ( n \to \infty ), we get:

[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{1}{(2n+3)(2n+2)} = 0 ]

Since the limit is less than 1, the series ( \frac{1}{(2n+1)!} ) converges absolutely by the Ratio Test.

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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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