上學期期末考 1.Let f:[1,∞)→R be nonnegative and decreasing and let n n dn = Σ f(k) - ∫f(x)dx , n belong to N k=1 1 Prove that {dn}n belong to N converges. 2.Test the following series for absoulte convergence ,conditional convergence , or divergence. (1) ∞ 1 (2) ∞ n^100 Σ ───── Σ ──── n=1 n^(2/3) n=1 n! (3)∞ 3 1 2 3 1 2 Σ ── - ── - ── + ── - ── - ── + ... n=1 √1 √2 √3 √4 √5 √6 3. x‧n^(3/2) Let fn(x) = ────── , n belong to N 1 + x‧n^2 Prove that {fn}n belong to N converges pointwise on [0,∞) and determine whether or not the convergence is uniform on [0,∞). How about on [0.001,∞)? 4.Let a,b belong to R with a < b and let fn be integrable on [a,b] for all n belong to N .Prove that if {fn}n belong to N converges uniformly on [a,b], then f is integrable on [a,b] . 5.(1)State the Weierstrass M-test . (2)Given a real-valued function that is continuous everywhere on R but differentiable nowhere on R . 6.(1)Find the Fourier series for the tunction f(x) = │x│, x belong to[-π,π] (2) ∞ 1 π Prove that Σ ───── = ── n=1 (2n-1)^2 8 7.Suppose that Sn is the n-th partial sum of the Fourier series of an integrable function f with period 2π .Prove that 1 π 1 π 1 π Sn(x) = ──‧∫f(t)Dn(x-t)dt = ──∫f(x-t)Dn(t)dt + ──∫f(x-t)Dn(t)dt π -π π 0 π 0 -- ┌─────┬──┬┬┬┬┬┬───┐ ┌──────────────────┤恭喜發財 └┐哈├┘└┘└┤爽就好│ │http://www.wretch.cc/blog/demonhell │ ┌─┘哈│Farewell└─┬─┘ └─────┬──────┬┬┬┬┬─┴┐ ┌─┴───┴───┬──┘ │你管我寫什麼├┼┼┼┤哞～├─┤不想被看，還是要擺│ └──────┴┴┴┴┴──┴─┴─────────┘ --

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