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    Sorry to bother, but how does the omission of the power of  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p} + \sqrt{q}} to the power of 4r-2 go away?

    Also, thanks for the reps as of late, and of course thanks for all these resources. They have been very useful for all my modules.

    http://www.madasmaths.com/archive/ma..._functions.pdf

    The very last question.
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    (Original post by AMarques)
    Sorry to bother, but how does the omission of the power of  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p} + \sqrt{q}} to the power of 4r-2 go away?

    Also, thanks for the reps as of late, and of course thanks for all these resources. They have been very useful for all my modules.

    http://www.madasmaths.com/archive/ma..._functions.pdf

    The very last question.
    Thing is, if I'm not mistaken, that series diverges.
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    (Original post by IrrationalRoot)
    Thing is, if I'm not mistaken, that series diverges.
    Hey,

    How is does it diverge? the expression  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} < 1 but  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} \rightarrow 1 as  p \rightarrow \infty and q is small.

    But considering only if they are real, positive, the "ratio" of this series is less than 1, so should converge? If I'm wrong, why would it diverge? Would love to know

    Edit: I don't really mean ratio, but I mean each term is essentially getting smaller and should tend to zero.
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    (Original post by AMarques)
    Hey,

    How is does it diverge? the expression  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} < 1 but  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} \rightarrow 1 as  p \rightarrow \infty and q is small.

    But considering only if they are real, positive, the "ratio" of this series is less than 1, so should converge? If I'm wrong, why would it diverge? Would love to know

    Edit: I don't really mean ratio, but I mean each term is essentially getting smaller and should tend to zero.
    All because the terms tend to 0 doesn't mean the sum converges.
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    (Original post by AMarques)
    Hey,

    How is does it diverge? the expression  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} < 1 but  \dfrac{ \sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} \rightarrow 1 as  p \rightarrow \infty and q is small.

    But considering only if they are real, positive, the "ratio" of this series is less than 1, so should converge? If I'm wrong, why would it diverge? Would love to know

    Edit: I don't really mean ratio, but I mean each term is essentially getting smaller and should tend to zero.
    The series is effectively 1+\dfrac{1}{3}+\dfrac{1}{5}+ \dfrac{1}{7}+\cdots which diverges. The rest of the stuff inside the summation is independent of r (so you can take it out as a constant).

    Not sure why you're considering p\rightarrow\infty, are we not looking at the same series?
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    (Original post by IrrationalRoot)
    The series is effectively 1+\dfrac{1}{3}+\dfrac{1}{5}+ \dfrac{1}{7}+\cdots which diverges. The rest of the stuff inside the summation is independent of r (so you can take it out as a constant).

    Not sure why you're considering p\rightarrow\infty, are we not looking at the same series?
    Hey,

    I understand what you mean, the series he gives would indeed diverge as the you could take out the two constants 2 and the expression above, leaving just  \frac{1}{2r-1} and hence giving the series you described, but, I'm more worried about this:
     \displaystyle \sum_{r=1}^{\infty} \dfrac{2}{2r-1} \bigg( \dfrac{\sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} \bigg)^{4r-2} = \ln\bigg({\dfrac{p + q}{2}}\bigg) - \dfrac{\ln{p}+\ln{q}}{2}
    Then went onto say that the the power gets removed.

    If the power was removed, I understand your point, I also recognise this series as a divergent one. But why was the power removed?

    I tried seeing a pattern when squaring this constant, but nothing came about.
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    (Original post by TeeEm)
    It does not disappear!!
    The text has a typo
    and my solution has omission.
    I will correct by tomorrow as I am very busy at present


    Please contact me in my website in the future as I do not wish to post in this site anymore
    Ok I shall do, and thanks for the response.
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    (Original post by AMarques)
    Hey,

    I understand what you mean, the series he gives would indeed diverge as the you could take out the two constants 2 and the expression above, leaving just  \frac{1}{2r-1} and hence giving the series you described, but, I'm more worried about this:
     \displaystyle \sum_{r=1}^{\infty} \dfrac{2}{2r-1} \bigg( \dfrac{\sqrt{p} - \sqrt{q}}{\sqrt{p}+\sqrt{q}} \bigg)^{4r-2} = \ln\bigg({\dfrac{p + q}{2}}\bigg) - \dfrac{\ln{p}+\ln{q}}{2}
    Then went onto say that the the power gets removed.

    If the power was removed, I understand your point, I also recognise this series as a divergent one. But why was the power removed?

    I tried seeing a pattern when squaring this constant, but nothing came about.
    Ah TeeEm has mentioned that the text has a typo. That explains.
 
 
 
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