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, pleasesolve 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] andn'=In+1 (x) ≤ In (x),and define sn [8, 2π – 8] → R, byShow thatR be given byLet fn [8, 2π8] → R be given byhn (x)Sn (x)Hint: Show that |sn(x)|Hint: Check that hn-n=1sincos (nx)n0 uniformly.Sn (x) =fn (2) = cos (nx)nΣfn (x).k=12sin () 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) sinsin cos ¹x.Show that there is M ≥ 0, such that, for all n € N and x € [8, 2π – 8]|Sn (x)| ≤ M.2x = [8, 2π - 8]Use part la) and 16) to show that hn converges uniformly on [8, 2π – 8].= ... =0, for alln=1fn 9n satisfies all the conditions of the Dirichlet's Test.

We shall use Dirichlets test as mentioned

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]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

I was doing my homework but look for an example but I don't know how they get the class midpoint
Please see the pictures. One contains the information needed and I need help with Part 4. The answer IS NOT 0.3173 OR 0.1507. please be certain of your work as I only have one attempt remaining.
Tanya takes her physics test and is told that her z-score is 3.10. What percentage of students scored below Tanya? The answer is 99.10% I would like to see the work for this problem please.
the average household income of Georgia is 89,679.25 and median household income for Georgia is 59,235 Is the median always lower than average? If not, please elaborate and provide an example that indicates opposite results.
A survey shows that 3 out of 4 of those drink coffee in a town with a population is 7900 adults how many of these adults would you expect to drink coffee?
Can someone explain the formulas needed and what variables go where? 445 randomly selected bulbs were tested in a laboratory, 316 lasted more than 500 hours. Find a point estimate of the proportion of all bulbs that last more than 500 hours.
I already have a,b,c could you help me find d,e,f,g?