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Math Help - Infinite Series Problems

  1. #1
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    Infinite Series Problems

    Hey guys, I have some infinite series problems I need my solutions checked to.

    Problem 1. Use the ratio test to determine whether the series converges or diverges. Explain how the test permits you to draw your conclusion.

    \Sigma^{\infty}_{n = 1}\frac{n^n}{n!}

    My Solution

    \lim_{n \to \infty}\frac{n^{n + 1}}{(n+1)n!}*\frac{n!}{n^n} = \lim_{n \to \infty}\frac{n}{n + 1} = 1

    Since \lim_{n \to \infty}\frac{\frac{n^{n + 1}}{(n + 1)!}}{\frac{n^n}{n!}} = 1, the ratio test fails.

    Here, I have trouble thinking my solution is right because the directions make no mention of the test failing. Is my solution correct?

    For problems 2 and 3 determine whether the series converges absolutely, converges conditionally, or diverges. Explain how you drew your conclusion in each case.

    Problem 2.

    \Sigma^{\infty}_{n = 1}(-1)^{n}\frac{e^n}{n!}

    Problem 3.

    \Sigma^{\infty}_{n = 1}(-1)^{n}\frac{1}{\sqrt{2n + 1}}

    I don't know how to do either of these problems, help please!

    Thanks in advance for all help!
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  2. #2
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    Problem one remember that (n+1)! is (n+1)n! so that cancels out the other n!.
    (N+1)^n+1 x n! = (n+1)^n+1/(n+1)..= [(n+1)/n]^n=e
    (n+1)! x N^n

    e is larger than 1 there for diverges
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  3. #3
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    For problem 2 and 3 use ratio test (n+1)/n and you will find out if it is absouletly convergent or not
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  4. #4
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    Quote Originally Posted by mturner07 View Post
    Hey guys, I have some infinite series problems I need my solutions checked to.

    Problem 1. Use the ratio test to determine whether the series converges or diverges. Explain how the test permits you to draw your conclusion.

    \Sigma^{\infty}_{n = 1}\frac{n^n}{n!}

    My Solution

    \lim_{n \to \infty}\frac{n^{n + 1}}{(n+1)n!}*\frac{n!}{n^n} = \lim_{n \to \infty}\frac{n}{n + 1} = 1

    Since \lim_{n \to \infty}\frac{\frac{n^{n + 1}}{(n + 1)!}}{\frac{n^n}{n!}} = 1, the ratio test fails.

    Here, I have trouble thinking my solution is right because the directions make no mention of the test failing. Is my solution correct?

    For problems 2 and 3 determine whether the series converges absolutely, converges conditionally, or diverges. Explain how you drew your conclusion in each case.

    Problem 2.

    \Sigma^{\infty}_{n = 1}(-1)^{n}\frac{e^n}{n!}

    Problem 3.

    \Sigma^{\infty}_{n = 1}(-1)^{n}\frac{1}{\sqrt{2n + 1}}

    I don't know how to do either of these problems, help please!

    Thanks in advance for all help!
    (1) correction ...

    \lim_{n \to \infty}\frac{\textcolor{red}{(n+1)}^{n + 1}}{(n+1)n!}*\frac{n!}{n^n}

    (2) use the ratio test to see if \sum \frac{e^n}{n!} converges ... if it does, it converges absolutely.

    (3) you should already know that \sum \frac{1}{\sqrt{2n+1}} diverges ... what do you know about the alternating series test?
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