(5) Consider the step function 1 if@ € (0, т], O if 0 0, f(0) = if 0 € (-T,0) -1 (a) Use Dini's criterion to show that the Fourier series of f converges to f pointwise on T (b) Show that sin((2k1)0) (2k1 SNf(0) k 0 Conclude that the alternating harmonic series over odd integers sums to ; that is, Σ T 2k1 k 0 4 (c) Prove that 2 (sin(2(N1)0) sin(0) (SNf(0) TT Conclude that lim (SNf'(0)= 0. N-o0 HINT: Use the formula for the N-th partial sum of the geometric series: N-1 1 - zN 1 z k-0

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
(5) Consider the step function
1 if@ € (0, т],
O if 0 0,
f(0) =
if 0 € (-T,0)
-1
(a) Use Dini's criterion to show that the Fourier series of f converges to f pointwise on T
(b) Show that
sin((2k1)0)
(2k1
SNf(0)
k 0
Conclude that the alternating harmonic series over odd integers sums to ; that is,
Σ
T
2k1
k 0
4
(c) Prove that
2 (sin(2(N1)0)
sin(0)
(SNf(0)
TT
Conclude that
lim (SNf'(0)= 0.
N-o0
Transcribed Image Text:(5) Consider the step function 1 if@ € (0, т], O if 0 0, f(0) = if 0 € (-T,0) -1 (a) Use Dini's criterion to show that the Fourier series of f converges to f pointwise on T (b) Show that sin((2k1)0) (2k1 SNf(0) k 0 Conclude that the alternating harmonic series over odd integers sums to ; that is, Σ T 2k1 k 0 4 (c) Prove that 2 (sin(2(N1)0) sin(0) (SNf(0) TT Conclude that lim (SNf'(0)= 0. N-o0
HINT: Use the formula for the N-th partial sum of the geometric series:
N-1
1 - zN
1 z
k-0
Transcribed Image Text:HINT: Use the formula for the N-th partial sum of the geometric series: N-1 1 - zN 1 z k-0
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 9 steps with 5 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,