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Thread: Calculus 2: Series/Sequences test problem

  1. #1
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    Calculus 2: Series/Sequences test problem

    So, i just took my exam. I think I did really well! Except, there were two questions that I sorta blanked out on and maybe was doing something wrong.

    1) The first one was find the convergence of ((3n^2+8)/(8n^2+3))^n. I used the root test and found that the limit < 1 which means that the series converges absolutely.

    However, the second part to this same series was whether the sequence ((3n^2+8)/(8n^2+3))^n converges or diverges? (if converges, compute lim as n-> inf of the seq).

    ---> so since the series converges, i was thinking that the sequence should converge (limit of seq. exists). However, when i tried to find the limit of the sequence, the sequence goes to inf (e^inf). Thus, i said that the sequence diverges to infinity.

    My Question: shouldn't the sequence converge if the series converge? Why couldn't i find a limit for the sequence then?



    2) Another question was simply write sin(5x)cos(5x) with maclaurin series using the trig identity : sin(2x) = 2sin(x)cos(x).

    I knew the sin(5x) and cos(5x) series with the memorized maclaurin series. However, i was having trouble rewriting sin(5x)cos(5x)... I thought it would be
    2sin(5x)cos(5x) = sin(10x) ---> sin(10x)/2.

    I probably rewrote it wrong with the trig identity. I completely blanked on how to use these trig identity!
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  2. #2
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    Re: Calculus 2: Series/Sequences test problem

    1) As $n$ gets large, you have essentially $\left(\dfrac{3}{8}\right)^n$. This limit approaches zero as $n \to \infty$.

    2) That is the correct trig identity.
    Thanks from lc99
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  3. #3
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    Re: Calculus 2: Series/Sequences test problem

    awh man. I knew something was wrong for problem 1...

    It just didn't make sense. I guess i computed the limit completely wrong!
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    Re: Calculus 2: Series/Sequences test problem

    Quote Originally Posted by lc99 View Post
    awh man. I knew something was wrong for problem 1...

    It just didn't make sense. I guess i computed the limit completely wrong!
    You just missed a minus sign. You should have gotten $e^{-\infty}$ as the limit, which is zero.
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