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**Unformatted text preview: **Math 334 Lecture #20 5.1: Power Series Once a fundamental set of solutions of the associated homogeneous ODE for a linear ODE with nonconstant coefficients like y 00 + xy + 2 y = g ( x ) is found, then a particular solution is found by Variation of Parameters. How are solutions of homogeneous linear ODEs with nonconstant coefficients found? By power series. Review of Power Series. A power series in x with center x is the infinite sum X n =0 a n ( x- x ) n where the a n s are real (or complex) numbers. Whenever infinitely many numbers are added together, there is always the question of whether the sum makes sense or not as a number. A power series converges (or makes sense as a number) at x if the sequence of partial sums has a limit: lim m m X n =0 a n ( x- x ) n exists . [Each partial sum is a finite sum which always makes sense.] A power series converges absolutely at x if lim m m X n =0 | a n ( x- x ) n | exists ....

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