Transcription of Chapter 7 Power series methods - Oklahoma State University ...
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Chapter 7 Power series methods - Oklahoma State University ...
Chapter 7. Power series methods Power series Note: 1 or lecture , in [EP], in [BD]. Many functions can be written in terms of a Power series . X. ak (x x0 )k . k=0. If we assume that a solution of a differential equation is written as a Power series , then perhaps we can use a method reminiscent of undetermined coefficients. That is, we will try to solve for the numbers ak . Before we can carry out this process, let us review some results and concepts about Power series . Definition As we said, a Power series is an expression such as . X. ak (x x0 )k = a0 + a1 (x x0 ) + a2 (x x0 )2 + a3 (x x0 )3 + , ( ). k=0. where a0 , a1 , a2 , .. , ak , .. and x0 are constants. Let n X. S n (x) = ak (x x0 )k = a0 + a1 (x x0 ) + a2 (x x0 )2 + a3 (x x0 )3 + + an (x x0 )n , k=0. denote the so-called partial sum. If for some x, the limit n X. lim S n (x) = lim ak (x x0 )k n n.
Power series methods 7.1 Power series Note: 1 or 1.5 lecture , §3.1 in [EP], §5.1 in [BD] Many functions can be written in terms of a power series X1 k=0 a k(x x 0)k: If we assume that a solution of a di erential equation is written as a power series, then perhaps we can use a method reminiscent of undetermined coe cients.
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