Solving Equations Involving Power Series

LogicalExpand[series1==series2]give the equations obtained by equating corresponding coefficients in the power series
Solve[series1==series2,{a1,a2,...}]solve for coefficients in power series

Solving equations involving power series.

Here is a power series.
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This gives an equation involving the power series.
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LogicalExpand generates a sequence of equations for each power of .
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This solves the equations for the coefficients a[i]. You can also feed equations involving power series directly to Solve.
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Some equations involving power series can also be solved using the InverseSeries function discussed in "Composition and Inversion of Power Series".

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