I Derivation for the indicial exponent in the Frobenius method

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I'm reading a book called Asymptotic Methods and Perturbation Theory, and I came across a derivation that I just couldn't follow. Maybe its simple and I am missing something. Equation 3.3.3b below. y(x) takes the form A(x)*(x-x0)^α and A(x) is expanded in a taylor series.

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Use \begin{split}<br /> \left(\sum_{n=0}^\infty a_nx^n \right)\left(\sum_{n=0}^\infty b_nx^n \right) &amp;= <br /> \sum_{n=0}^\infty \sum_{m=0}^\infty a_n b_m x^{n+m} \\<br /> &amp;=\sum_{k=0}^\infty \left(\sum_{n=0}^n a_n b_{k-n}\right) x^k. \end{split}
 
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Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
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