How do you use the limit comparison test to test if #1/(n!)# is convergent?

To test the convergence of the series (\frac{1}{{n!}}) using the limit comparison test, we compare it with a known convergent series. One commonly used convergent series for comparison is the geometric series.

Let's consider the geometric series (b_n = \frac{1}{{2^n}}), which converges since its common ratio is less than 1.

Now, we form the limit:

[\lim_{{n \to \infty}} \frac{a_n}{b_n} = \lim_{{n \to \infty}} \frac{\frac{1}{{n!}}}{\frac{1}{{2^n}}}]

This simplifies to:

[\lim_{{n \to \infty}} \frac{2^n}{{n!}}]

We can apply the ratio test to find this limit:

[\lim_{{n \to \infty}} \frac{2^{n+1}}{{(n+1)!}} \cdot \frac{{n!}}{{2^n}}]

After simplifying, this becomes:

[\lim_{{n \to \infty}} \frac{2}{n+1}]

As (n) approaches infinity, the limit goes to zero. Since the limit is finite and positive, the series (\frac{1}{{n!}}) converges by the limit comparison test with the convergent series (\frac{1}{{2^n}}).