# Derivative of (-1)^n/l'hopital's rule

1. Nov 26, 2008

### vinnie

1. The problem statement, all variables and given/known data

I am tryint to show that a series converges using the alternating series test. First, we show that 0<an+1<an and then that the limit of the expression equals 0. (the question says show that the series converges using the alternating series test, so I know it converges.)

2. Relevant equations

l'hopital's rule,

3. The attempt at a solution

the series is ((-1)^n)/(2sqrt(3n))

I need to use l'hopital's to show that the limit equals zero. The problem is, what is the derivative of (-1)^n if ln (-1) doesn't exist?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 26, 2008

### Unco

Hi Vinnie,

If a_n is the absolute value of the nth term (i.e., the factor being multiplied by (-1)^n), in this case 1/(2*sqrt(3n)), then part of the alternating series test asks that $$\lim_{n\to \infty} a_n = 0$$, so (-1)^n does not occur in the limit.

3. Nov 26, 2008

### vinnie

so we would take the limit of 1/2srqrt(3n)?

how would we find the sum of this infinite series?

4. Nov 26, 2008

### Unco

For the second part of alternating series test, yes. The other part of the test, as you have said, requires you to show that the sequence 1/(2*sqrt(3n)) (a_n as I defined previously) is decreasing. Review your notes.

The alternating series test tells you the series converges, which is all the question required. This particular series' sum cannot be written in elementary terms. In mathematics, determining precisely what a quantity is may be a completely different task to simply showing that the quantity exists.

5. Nov 26, 2008

### moo5003

Step 1: Determine a_n
Step 2: Determine if the SEQUENCE a_n is monotone decreasing

ie: The limit is 0 and it is infact decreasing a_n < a_(n-1)

Wikipedia should have more info on the alternating series test if this is not clear enough.

Note: Not sure why you want to use l'hospital, its not applicable.

6. Nov 26, 2008

### vinnie

Now that I know about (-1)^n, I know it's not applicable.

7. Nov 26, 2008

### vinnie

I was speaking of the second part of the test. I know all about the first part. Thank you.

8. Nov 26, 2008

### Dick

To show 1/(2sqrt(3n)) is decreasing, you could take the derivative of 1/(2sqrt(3x)) and show that it is negative for positive x, or just note that since n is increasing 1/sqrt(n) must be decreasing.