# Homework Help: Comparison test for convergence problem: why is this incorrect?

1. Feb 27, 2012

### skyturnred

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

The original question is posted on my online-assignment. It asks the following:

Determine whether the following series converges or diverges:

$\sum^{\infty}_{n=1}$$\frac{3^{n}}{3+7^{n}}$

There are 3 entry fields for this question. One right next to the series above with the following options:

either $\succ$ or $\prec$ then next to that there is a field in which to input the thing that I am going to compare it to.

the third and final field I choose divergent or convergent

2. Relevant equations

3. The attempt at a solution

So I compared it to ($\frac{4^{n}}{6^{n}}$) because the original series is clearly less than this one. By doing the comparison test I determined that the series converges.

So I get "less than" right and "converges" right but I didn't get the other part right. Isn't it true that the original series is less than the one I decided above? Or was I wrong somewhere else?

2. Feb 27, 2012

### lanedance

Everything you said sounds trues, though I'm not sure I understand the question?

Also, why wouldn't you just compare to $\sum_n \frac{3^n}{7^n}$?

Last edited: Feb 27, 2012
3. Feb 27, 2012

### skyturnred

That must be it.. I was just slight confused as to whether it was bigger or smaller than the original series, so I wanted to be EXTRA sure by choosing the one I mentioned above. I guess that must have been where I went wrong. But still, is what I did above correct? I realize that I could have made an "easier" decision for what to compare it to, but isn't what I chose still correct?

Thanks!

4. Feb 27, 2012

### lanedance

yeah looks ok to me, generally want to choose the easiest to compare and the closest possible.

For the pupose of the convergence, as long as for some n>N each term yn> xn and (sum yn) converges, then (sum xn) converges

5. Feb 27, 2012

### kai_sikorski

If it's an automatically graded online assignment the software might be hard-coded to only accept the most natural choice. What you did is correct though.