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Math Help - does this converge?

  1. #1
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    does this converge?

    Sumation from 1 to infinity of [3^n + 4] / 2^n converges or diverges?
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  2. #2
    Junior Member roy_zhang's Avatar
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    Given \sum\limits_{n=1}^\infty \frac{3^n+4}{2^n}, I guess you can apply direct comparison test here.

    Let b_n=\frac{3^n+4}{2^n}, consider another sequence a_n=\frac{3^n}{2^n}. We have 0\le a_n\le b_n,\;\;\forall n. Since the geometric series \sum\limits_{n=1}^\infty a_n=\sum\limits_{n=1}^\infty (\frac{3}{2})^n is obvious divergent. We can conclude that the given series \sum\limits_{n=1}^\infty \frac{3^n+4}{2^n} must also be divergent.
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  3. #3
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    \sum_{n=1}^{\infty} \frac{3^n + 4}{2^n} Diverges, you can use the comparison test to show this, using b_n \equiv \sum_{n=1}^{\infty} \frac{3^n}{2^n}=(\frac{3}{2})^n then using the root test on b_n, you get \sqrt[n]{(\frac{3}{2})^n}  = \frac{3}{2} > 1, therefore it diverges
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