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Math Help - Series and Sequences ex

  1. #1
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    Series and Sequences ex

    Hello,

    I'm preparing for my mathematics exams and am a little bit struggling with series and functions. Since my closest lecturer is 3000km away, decided to ask some help online

    The following Series are convergent or divergent?
    http://img69.imageshack.us/img69/922...otestexam2.jpg
    http://img231.imageshack.us/img231/2...iotestexam.jpg
    I solved it but really not sure if it is right.

    And was struggling to solve this one for Sequences: whether or not this sequence converge, if does state limit: 2010^n / n^2010.
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  2. #2
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    The second one is correct. Now for the first one

    if a_n = \frac{(-1)n}{\sqrt{n}} = -\sqrt{n} , then it does not converge since the term oscillates more an more wildly.

    If a_n = \frac{(-1)^n}{\sqrt{n}}, then you can just use the alternating series test. This term goes to 0 as n goes to infinity, so we have that the series converges! The work you have shown is incorrect though, because you cannot factor -1 out of the absolute value. The limit is indeed 1, so it gives no information on convergence.
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  3. #3
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    Thank You for the explanation and sorry for confusing you, it was (-1)^n in the example - bad paint skills!

    Maybe have any ideas on the bottom exercise, the sequence 2010^n / n^2010 diverge ir converge ( limit ? )
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  4. #4
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    The numerator is an exponential while the denominator is a polynomial. At first this ratio is very small for small numbers. However, when n = 2010, the ratio is 1. After that the numerator wins.

    If you compare consecutive terms you have:

    \displaystyle \frac{a_{n+1}}{a_n} = \frac{2010^{n+1}}{(n+1)^{2010}}\frac{n^{2010}}{201  0^n} = 2010\left(\frac{n}{n+1}\right)^{2010}

    The limit of this is of course 2010 and since it's much larger than 1, the sequence diverges to infinity.
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