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Math Help - Help with the first three terms of this infinite series please

  1. #1
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    Help with the first three terms of this infinite series please

    Hi,

    I'd really appreciate some help figuring out the terms of this infinite sum:

    \sum_{n=0}^{\infty }c_n

    where

    c_0=k=\sin\theta

    k'\equiv \cos\theta

    k_{n+1}=\frac{1-k'_n}{1+k'_n}


    Would the first three terms be...

    \sin\theta+\frac{1-\cos(\arcsin\theta)}{1+\cos(\arcsin\theta)}+\frac{  \left (1-\cos\left ( \arcsin\left ( \frac{1-\cos(\arcsin\theta)}{1+\cos(\arcsin\theta)} \right )\right ) \right )}{\left (1+\cos\left ( \arcsin\left ( \frac{1-\cos(\arcsin\theta)}{1+\cos(\arcsin\theta)} \right )\right ) \right )}

    ?


    Thanks
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  2. #2
    MHF Contributor ebaines's Avatar
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    I don't see why you have arcsin in your answer. You have:

    <br />
k_0 = sin \theta<br />

    <br />
k_0' = cos \theta<br />

    <br />
k_1 = \frac {1- k_0'} {1+ k_o'} = \frac {1+cos \theta} {1- cos\theta}<br />

    <br />
k'_1 = \frac {2 sin \theta} {(1-cos \theta)^2}<br />

    <br />
k_2 = \frac {1+ \frac {2 sin \theta} {(1-cos \theta)^2}} {1 - \frac {2 sin \theta} {(1-cos \theta)^2}} = \frac {1 - 2 cos \theta + 2 sin \theta + cos^2 \theta} {1 - 2 cos\theta -2 sin \theta + cos^2 \theta}<br />
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