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How to figure out $$\sum_{n = 1}^{\infty}\frac{(-1)^nn!2^n}{1\times3\times\cdots\times(2n-1)}$$ is convergent or divergent? I tried to use ratio test, but the limit is 1 which means it is inconclusive. What else test should I use?

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I reckon that $n!2^n = 2\times4\times6\times\cdots\times 2n$ is the product of the first $n$ even numbers, and that is bigger than the product of the first $n$ odd numbers.

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Hint: If the general term of a series does not converge to $0$, then the series does not converge.

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