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Math Help - limit question on a limit comparison test problem.

  1. #1
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    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.


    solution.
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  2. #2
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    Quote Originally Posted by Evan.Kimia View Post
    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.
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