(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
(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
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fa0477a-e18e-4d9b-9d22-f38e779fb633%2F3348b5a2-08d0-415e-9c8c-956271ea0b8b%2Fnj89iw5.png&w=3840&q=75)
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
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
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