solutions for quiz xxix

# 2 1 2 2 x2 2 x 1 4 2 x4 2 x3 1 6 2 x6

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Unformatted text preview: roved that the series (∗) is convergent for an arbitrary real number u. It is then clear that the original series too is convergent for an arbitrary real number x. 5 ⋆ Note Jx = : Let’s duplicate (2b): 1 0!! 2 − 1 2!! 2 x2 + 2 x+ 1 4!! 2 x4 − 2 x3 − 1 6!! 2 x6 + 2 x5 + 1 2 x8 − 2 x7 − 8!! 1 10!! 2 x10 + ··· . 2 x9 + ··· , Diﬀerentiate this twice: J′ x = − 2 2!! 4 4!! 6 6!! 8 8!! 10 10!! and J ′′ x = − 2 ·1 2!! 2 + 4 ·3 4!! 2 x2 − 6 ·5 6!! 2 x4 + 8 ·7 8!! 2 x6 − 10 ·9 10!! 2 x8 + 12 ·11 12!! 2 x10 − ···. J ′′ x can be written as follows: J ′′ x =− 1 2 0!! 2 + 3 4 2!! 2 x2 − 5 6 4!! 2 x4 + 7 8 6!! 2 x6 − 9 10 8!! 2 x8 + 11 12 10!! 2 x10 − ··· . 2 x10 + ··· . 2 x10 − ··· . Add up J x and J ′′ x : Jx J ′′ x = =− 1 0!! 2 1 2 0!! 2 − + 1 2!! 2 3 4 2!! x2 + 2 2 x− 1 4!! 2 5 6 4!! x4 − 4 2 x+ 1 6!! 2 7 8 6!! x6 + 6...
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## This note was uploaded on 01/12/2014 for the course MATH 116 taught by Professor Scholle,minho during the Fall '08 term at Kansas.

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