# limit question on a limit comparison test problem.

• August 8th 2010, 10:37 AM
Evan.Kimia
limit question on a limit comparison test problem.
I understand why the limit comparison test is being used on this problem, but im not sure how they are simplifying B sub n into A sub n. i would of thought it would simplify to ((1+2^n)*(3^n))/((1+3^n)(2^n). Thanks.

problem.
http://i36.tinypic.com/24diqkk.jpg

solution.
http://i38.tinypic.com/14ocabo.jpg
• August 8th 2010, 11:29 AM
General
Quote:

Originally Posted by Evan.Kimia
I would of thought it would simplify to ((1+2^n)*(3^n))/((1+3^n)(2^n)

Correct.
Take 2^n as common factor in the numerator, and 3^n as a common factor in the denominator. Cancel the same guys, and you will be fine.

There is another solution using the Basic Comparison Test:

$\dfrac{1+2^n}{1+3^n} \leq \dfrac{1+2^n}{3^n} \leq \dfrac{2^n+2^n}{3^n} = 2 \cdot \left( \dfrac{2}{3} \right)^n$

and the last one is a convergent geometric series.