solutions for quiz xxix

# r work consider the series with variable u 0 a

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Unformatted text preview: 1 3! 2ℓ 1 2 · 2 · x2 ℓ . x 2 6 + 1 4! 2 x 2 8 − ··· . ∞ (3) 1 ℓ The radius of convergence for −1 ℓ=0 · x2 ℓ 2 2 ℓ !! R = +∞. Work : Consider the series with variable u: ∞ ℓ=0 aℓ = 2 · uℓ . 2 ℓ !! 1 ℓ Set 1 ℓ −1 −1 . 2 Then 2 ℓ !! aℓ 1 = 2 , 2 ℓ !! aℓ+1 1 = 2 2 ℓ + 2 !! . 1 2 2 ℓ !! =⇒ lim ℓ− ∞ → aℓ aℓ+1 = lim ℓ− ∞ → 1 2 2 ℓ + 2 !! 4 is = lim ℓ− ∞ → 2 2 ℓ + 2 !! 2 ℓ !! 2 = lim ℓ− ∞ → 2ℓ + 2 = + ∞. Hence the series ∞ ( ∗) ℓ=0 1 ℓ −1 2 · uℓ 2 ℓ !! with variable u has radius of convergence + ∞. Then the original series ∞ ℓ=0 1 ℓ −1 2 · x2 ℓ 2 ℓ !! with variable x must have radius of convergence + ∞. Indeed, the original series with variable x is merely the outcome of the substitution of u = x2 in the series (∗) above with variable u . We have just p...
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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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