Math 321 - The Dirichlet Test

For each xed x lim gn1 x 0 so 1 guarantees that sn

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Unformatted text preview: (x) ≥ 0, gn+1 (x) ≥ 0 and every gk (x) − gk+1 (x) ≥ 0. For each fixed x, lim gn+1 (x) = 0. So (1) guarantees that {sn (x)} is a Cauchy sequence and hence conn→∞ verges. Call the limit s(x). Taking the limit of (1) as m → ∞ gives s(x) − sn (x) ≤ 2M gn+1 (x) February 3, 2008 The Dirichlet Test 1 Since gn+1 (x) converges uniformly to zero as n → ∞, we have that sn (x) converges uniformly to s(x) as n → ∞. n ∞ ∞ n 1z z , n=0 n R and Example. We shall consider three different power series: n=0 R ∞ 1 zn , for some fixed R > 0. For all three series, the radius of convergence is n=0 n2 R exactly R since, for ℓ ∈ {0, 1, 2}, ℓ n lim sup n→∞ 11 nℓ Rn = 1 R n lim sup n→∞ 1 n = 1 R...
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