notes 5-1

# notes 5-1 - term carefully because the general form for the...

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SECTION 5.1 REVIEW OF POWER SERIES X n =0 a n ( x - x 0 ) n This power series converges at z if lim p →∞ p X n =0 a n ( z - x 0 ) n exists, which means that it is a real number. If the power series does not converge at z , then it diverges at z . Every power series has a radius of convergence ρ so that the power series converges (absolutely) at z when | z - x 0 | < ρ and diverges at z when | z - x 0 | > ρ . Our diﬀerential equations textbook says that we then take the interval of convergence to be x 0 - ρ < z < x 0 + ρ. Special cases are 1. ρ = 0 2. ρ = 3. the endpoints of the interval. Inside the interval of convergence a power series deﬁnes a function f ( z ) = X n =0 a n ( z - x 0 ) n . Such functions are called analytic . Inside the interval of convergence such functions can be added, subtracted, multiplied by constants, diﬀerentiated, and integrated by carrying out the operation on the series term-by-term. When you diﬀerentiate, you must handle the constant

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Unformatted text preview: term carefully, because the general form for the term often does not work well. Analytic functions can even be multiplied, but you must treat the power series as polynomials, so lots of like terms must be combined. It’s a mess. They can even be divided when the denominator isn’t zero, but that’s an even bigger mess. EXAMPLES. (1) ∞ X n =0 x n n ! ! = (2) ∞ X j =0 (-1) j x 2 j (2 j )! = (3) For f ( z ) = ∞ X n =0 a n ( z-x ) n , ﬁnd f (4) ( x ). (4) Rewrite the power series ∞ X n =2 n ( n-1) a n x n-2 as a power series whose generic or general term involves x n . THERE IS NO HOMEWORK FOR SECTION 5.1 – BUT PLEASE MAKE SURE YOU REVIEW THAT MATERIAL....
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notes 5-1 - term carefully because the general form for the...

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