# How do you determine if series #1/n!# converge or diverge with comparison test?

You first prove by induction that

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To determine if the series ( \sum \frac{1}{n!} ) converges or diverges using the comparison test, we compare it to a known convergent or divergent series. In this case, since ( n! ) grows much faster than ( n^k ) for any positive integer ( k ), we can compare it to the series ( \sum \frac{1}{n^2} ), which is a convergent p-series with ( p = 2 ).

Therefore, we can establish the following comparison: [ 0 < \frac{1}{n!} < \frac{1}{n^2} ]

Since the series ( \sum \frac{1}{n^2} ) converges, and ( \frac{1}{n!} ) is smaller than ( \frac{1}{n^2} ) for all ( n ), by the comparison test, the series ( \sum \frac{1}{n!} ) also converges.

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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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