solutions for quiz xxix

Hence we obtain x j x j x xj x 0 bessels dierential

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Unformatted text preview: 2 x− 1 8!! 2 9 10 8!! x8 − 8 2 x+ 1 10!! 11 12 10!! + J x + J ′′ x = 1 2 0!! 2 − 1 4 2!! 2 x2 + 1 6 4!! 2 x4 − 6 1 8 6!! 2 x6 + 1 10 8!! 2 x8 − 1 12 10!! 2 x10 + ··· . Realize that x multipled to this exactly equals −J ′ x . Hence we obtain x J ′′ x + J′ x + xJ x =0 Bessel’s differential equation . J x is called the Bessel function of the first kind . n2 ∞ [II] (10pts) (1) The radius of convergence for 1− n=1 1 n xn is R = e. n2 Work : Set aℓ = =⇒ R = 1− 1 n . Then n2 an an+1 = = 1− 1 n 1− 1 n+1 , (n+1)2 . n2 lim n− ∞ → an an+1 = lim n− ∞ → In order to compute this limit, take the logarithm: 7 1− 1− 1 n 1 n+1 (n+1)2 . n2 1− ln R = lim ln n− ∞ → 1− = = = = = + 1 n lim n− ∞ → lim n− ∞ → lim n− ∞ → lim n− ∞ → lim n− ∞ → 2 n ln 2 n ln 2 n ln 2 n ln 2 n n2 + 2n + 1 1 n+1 1− 1− 1− 1− − 1 n (n+1)2 − 1 n − 1 n + 1 n + n+1 n+1 n+1 n+1 2 2 2 2 ln ln ln ln 1− 1 n+1 n n+1 1 n n+1 n 1+ 1 1 1 1 1 − − − − − ·...
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