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Math Help - How do I know this series decreases or not?

  1. #1
    s3a
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    How do I know this series decreases or not?

    We're supposed to use the alternating series to say if this diverges or converges and if we do not know what it does using the alternating series test (regardless of what another test says) then we need to state that as well.
    However, the answer to the question is that it converges and since the question wants us to only use the alternating series test. Here is what confuses me: The limit DOES go to zero which is good but it does not seem to be decreasing! I tried b_n and b_(n+1) and it seems to increase instead. Am I doing something wrong?

    Any input would be greatly appreciated!
    Thanks in advance!
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    Quote Originally Posted by s3a View Post
    We're supposed to use the alternating series to say if this diverges or converges and if we do not know what it does using the alternating series test (regardless of what another test says) then we need to state that as well.
    However, the answer to the question is that it converges and since the question wants us to only use the alternating series test. Here is what confuses me: The limit DOES go to zero which is good but it does not seem to be decreasing! I tried b_n and b_(n+1) and it seems to increase instead. Am I doing something wrong?

    Any input would be greatly appreciated!
    Thanks in advance!
    Why wouldn't it decrease? The denominator is greater the numerator.
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  3. #3
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    The faster way is to use the derivative, put f(x)=\frac{x^3+1}{x^4+1}, now does f'(x)<0 ? If so, then a_n=\frac{n^3+1}{n^4+1} is strictly decreasing.
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  4. #4
    s3a
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    The derivative leads to n^6 + 4n^3 - 3 = 0 and therefore n > 0 which means that it is increasing, doesn't it?
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    Quote Originally Posted by s3a View Post
    The derivative leads to n^6 + 4n^3 - 3 = 0 and therefore n > 0 which means that it is increasing, doesn't it?
    No, you have the signs backward. The derivative of \frac{x^3+ 1}{x^4+ 1} is \frac{(3x^2)(x^4+ 1)- (4x^3)(x^3+ 1)}{(x^4+ 1)^2}= \frac{3x^6+ 3x^2- 4x^6- 4x^3}{(x^4+1)^2}= \frac{-x^6- 4x^3+ 3x^2}{(x^4+1)^2}
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    Quote Originally Posted by dwsmith View Post
    Why wouldn't it decrease? The denominator is greater the numerator.
    The fact that the denominator is greater than the numerator means that each fraction is less than 1, not that they are decreasing.

    If you meant that the degree of the denominator is greater than the degree of the numerator, then, yes, that implies that they are eventually decreasing.
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