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Math Help - Ratio and Root Test

  1. #1
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    Ratio and Root Test

    How would you work out if these series converge or diverge?

    Problem 1

    n=1 to infinity n!/10^n

    Problem 2

    n=1 to infinity ((n-2)/n)^n

    Problem 3

    n=1 to infinity (-2)^n/3^n
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  2. #2
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    What exactly are you having troubles with? You are already told which tests to use.

    For example, the first one. Let a_n = \frac{n!}{10^n}.

    Then: \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left| \frac{(n+1)!}{10^{n+1}} \cdot \frac{10^n}{n!}\right| = \lim_{n \to \infty} \left|\frac{(n+1)n!}{10^n \cdot 10} \cdot \frac{10^n}{n!}\right| = \cdots
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  3. #3
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    still not sure how to do the third problem though....can't use root test because of the negative term. There too many tests....
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  4. #4
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    Quote Originally Posted by TAG16 View Post
    still not sure how to do the third problem though....can't use root test because of the negative term. There too many tests....
    Problem 3

    n=1 to infinity (-2)^n/3^n
    \sum_{n=1}^{\infty} \frac{(-2)^n}{3^n} = \sum_{n=1}^{\infty} \left(\frac{-2}{3}\right)^n

    geometric series with |r| < 1 ???
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