Test the series for convergence

**boquist** · August 15th 2010, 04:05 PM

I am struggling through this new material.
Here is the problem:

Test the series for convergence-photo-5.jpg

I know from the limit comparison test that both b_n and a_n either converge or diverge, but I cant seem to find out which of the two it is...

How can I find whether b_n is convergent or divergent??

Thank you for the help!

**skeeter** · August 15th 2010, 04:29 PM

first, note that ...

$\displaystyle \frac{1 + 2^n}{1 + 3^n} = \frac{1}{1+3^n} + \frac{2^n}{1+3^n}$

individually ...

$\displaystyle \frac{1}{1+3^n} < \frac{1}{3^n} = \left(\frac{1}{3}\right)^n$

$\displaystyle \frac{2^n}{1+3^n} < \frac{2^n}{3^n} = \left(\frac{2}{3}\right)^n$

... you should be able to figure it out from this point.

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