Page 98 - Calculus for the Clueless, Calc II

Example 25—

Examine !

Note

6! means 6(5)(4)(3)(2)(1).

Also Note:

(k + 1)! = (k + 1)(k!). That is, 10! = 10(9!), and so on.

Let us again use the ratio test.

The series converges.

Example 26—

Let's now look at !

This is a little trickier than most. Again, we use the ratio test.

Since the ! diverges.

Example 27—

In order to show the third part of the previous theorem, you should apply the ratio test to both the harmonic
series and the p 2 series. Both give a ratio of 1. The first series diverges, and the second converges. So, if the
ratio is 1, we must indeed use another test.

Test 6 (Root Test)

Given . Take . If r > 1, it diverges. If r < 1, it converges. If r = 1, use another
test.

Note