# Calculus II - Alternating Series Test - Convergent?

1. Apr 2, 2012

### captcouch

Hello! I was working some practice problems for a Calc II quiz for Friday on the alternating series test for convergence or divergence of a series. I ran into a problem when I was working the following series, trying to determine whether it was convergent or divergent:

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

$\sum$ (-1)n$\frac{3n-1}{2n+1}$
n=1

2. Relevant equations
Alternating Series Test:
For an alternating series

$\sum$ (-1)n-1bn
n=1

to converge, it must satisfy the following conditions:

1. bn+1 ≤ bn for all n in the series
2. lim bn = 0
n$\rightarrow$∞

Test for Divergence of a Series:
If the limit as n approaches infinity of a series does not exist or does not equal 0, the series is divergent.

3. The attempt at a solution
I first began by analyzing the limit as n approaches infinity for the series. By dividing the coefficients of the highest exponential power of the top and bottom (in this case, 1), I found the limit of the series to be $\frac{3}{2}$. It did not satisfy the second condition of the alternating series test, and as such, I sought to double-check with the Test for Divergence of a Series.

(-1)n's limit does not exist, as it is always alternating back and forth between -1 and 1.

$\frac{3n-1}{2n+1}$'s limit, as mentioned above, I found to be $\frac{3}{2}$.

From this information, I concluded that the limit did not exist, and that the series was divergent.

I decided to check the answer from the back of the textbook, and it said it was convergent! I ran through it again a couple of times and got the same result: divergent. I'm not sure what I did wrong here - could I please have some insight from someone else to shed some light on the situation? Thank you very much!

Last edited: Apr 2, 2012
2. Apr 2, 2012

### RoshanBBQ

I'll tell you what's wrong here: your book. That series is definitely not convergent.

3. Apr 2, 2012

### captcouch

Man, you've got to be kidding me. Nobody's perfect, I guess. I'll be sure to tell my professor in the morning. Thanks for the help!

4. Apr 3, 2012

### HallsofIvy

Staff Emeritus
The most fundamental theorem about infinite series, typically the first proved in a textbook, is
If $\lim_{n\to\infty} a_n\ne 0$, then $\sum a_n$ does NOT converge.