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1. la. Let 80 and hn [8, 2π-8] →R be given by cos (nx) 1b.. = Use Dirichlet's Test to show that the series hn converges uniformly on [8, 278]. That is, please n=1 solve the following problems: Let gn (x) x € [8, 278] and and define sn [8, 27-6] → R, by Show that for all n E N and x = [8, 27 - 8]. Hint: Use the definition to show that 1/1 → 0 uniformly. Let fn [8, 278] R be given by hn (x) x x [8, 2π8]. Show that gn In+1 (x) ≤ 9n (x), Sn (x) = = Hint: Show that sn(x)| ≤ sin §. n Sn (x) fn (2) = cos (nx) = - n Σfn (x). k=1 sin (1) cos ((n+¹)x) sin (2) 2 Hint: Use the identities sin a-sin 3 = 2 cos at sin One can show that sin() Sn = Ek-1 cos(kx) sin Show that there is M≥ 0, such that, for all n |Sn (x)| ≤ M. g uniformly, where g(x) = 0, for all x € [8, 2π - 8] and cos a sin 3 = (sin(a + 3) - sin(a− 3)). == sin COS n+1x. N and x = [8, 27-8]

1. la. Let 80 and hn [8, 2π-8] →R be given by cos (nx) 1b.. = Use Dirichlet's Test to show that the series hn converges uniformly on [8, 278]. That is, please n=1 solve the following problems: Let gn (x) x € [8, 278] and and define sn [8, 27-6] → R, by Show that for all n E N and x = [8, 27 - 8]. Hint: Use the definition to show that 1/1 → 0 uniformly. Let fn [8, 278] R be given by hn (x) x x [8, 2π8]. Show that gn In+1 (x) ≤ 9n (x), Sn (x) = = Hint: Show that sn(x)| ≤ sin §. n Sn (x) fn (2) = cos (nx) = - n Σfn (x). k=1 sin (1) cos ((n+¹)x) sin (2) 2 Hint: Use the identities sin a-sin 3 = 2 cos at sin One can show that sin() Sn = Ek-1 cos(kx) sin Show that there is M≥ 0, such that, for all n |Sn (x)| ≤ M. g uniformly, where g(x) = 0, for all x € [8, 2π - 8] and cos a sin 3 = (sin(a + 3) - sin(a− 3)). == sin COS n+1x. N and x = [8, 27-8]

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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Please answer the question, with Dirichlet's Test, It is a practice question for an exam, THe various steps and hints have been provided also, Thanks

1. Let 8 >0 and hn: [8, 2π – 8] - 1c) 1b. . Use Dirichlet's Test to show that the series hn converges uniformly on [8, 2π – 6]. That is, please solve the following problems: for all n E N and x € [8, 2π – 6]. Hint: Use the definition to show that 1/1 → 1a. Let 9n (x) = 1/1, x = [6, 2π – 6]. Show that 9n → g uniformly, where g(x) = x € [8, 2π - 8] and n' = In+1 (x) ≤ In (x), and define sn [8, 2π – 8] → R, by Show that R be given by Let fn [8, 2π8] → R be given by hn (x) Sn (x) Hint: Show that |sn(x)| Hint: Check that hn - n=1 sin cos (nx) n 0 uniformly. Sn (x) = fn (2) = cos (nx) n Σfn (x). k=1 2 sin () cos (+¹)x) sin (2) Hint: Use the identities sin a-sin ß = 2 cos + sin a and cos a sin ß = 1/2 (sin(a + 3) - sin(a − 3)). One can show that sin() sn = Σk-1 cos(kx) sin sin cos ¹x. Show that there is M ≥ 0, such that, for all n € N and x € [8, 2π – 8] |Sn (x)| ≤ M. 2 x = [8, 2π - 8] Use part la) and 16) to show that hn converges uniformly on [8, 2π – 8]. = ... = 0, for all n=1 fn 9n satisfies all the conditions of the Dirichlet's Test.
Transcribed Image Text:1. Let 8 >0 and hn: [8, 2π – 8] - 1c) 1b. . Use Dirichlet's Test to show that the series hn converges uniformly on [8, 2π – 6]. That is, please solve the following problems: for all n E N and x € [8, 2π – 6]. Hint: Use the definition to show that 1/1 → 1a. Let 9n (x) = 1/1, x = [6, 2π – 6]. Show that 9n → g uniformly, where g(x) = x € [8, 2π - 8] and n' = In+1 (x) ≤ In (x), and define sn [8, 2π – 8] → R, by Show that R be given by Let fn [8, 2π8] → R be given by hn (x) Sn (x) Hint: Show that |sn(x)| Hint: Check that hn - n=1 sin cos (nx) n 0 uniformly. Sn (x) = fn (2) = cos (nx) n Σfn (x). k=1 2 sin () cos (+¹)x) sin (2) Hint: Use the identities sin a-sin ß = 2 cos + sin a and cos a sin ß = 1/2 (sin(a + 3) - sin(a − 3)). One can show that sin() sn = Σk-1 cos(kx) sin sin cos ¹x. Show that there is M ≥ 0, such that, for all n € N and x € [8, 2π – 8] |Sn (x)| ≤ M. 2 x = [8, 2π - 8] Use part la) and 16) to show that hn converges uniformly on [8, 2π – 8]. = ... = 0, for all n=1 fn 9n satisfies all the conditions of the Dirichlet's Test.
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