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.
solution.
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.
solution.
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:
$\displaystyle \dfrac{1+2^n}{1+3^n} \leq \dfrac{1+2^n}{3^n} \leq \dfrac{2^n+2^n}{3^n} = 2 \cdot \left( \dfrac{2}{3} \right)^n$
and the last one is a convergent geometric series.