Question 4. Define the power series * 2k+1 2k +1 r2k+1 T(1):= L (2k + 1)² S(1) := k-0 (a) Show that lim log(1 – r)S(1 – a) = 0 and lim log(r)S(r) = 0. (Hint: For the first one, recall from $3.4.1 of lecture notes that r| < = |log(1 + )| < 2|. Use L'Hôpital's rule (more than once if necessary). For the second one show that re (0, 1) Given e e (0, ), show that (꾸s (2)s (b) log (a-)dr = - log ((1 – e)²) S(1 –e) + log (e²) S(e) + 2(T(1- e) – T(e)). r² – 1 (Hint: Use the formula for the geometric sum on . Later use integration by parts.) (c) Show that 1-« log (r²) dr = r2 -1 1 22 72k + 1)² lim k-0 (2k + 1)² ° (Hint: Use Abel's theorem to get uniform convergence of T on [0, 1].)
Question 4. Define the power series * 2k+1 2k +1 r2k+1 T(1):= L (2k + 1)² S(1) := k-0 (a) Show that lim log(1 – r)S(1 – a) = 0 and lim log(r)S(r) = 0. (Hint: For the first one, recall from $3.4.1 of lecture notes that r| < = |log(1 + )| < 2|. Use L'Hôpital's rule (more than once if necessary). For the second one show that re (0, 1) Given e e (0, ), show that (꾸s (2)s (b) log (a-)dr = - log ((1 – e)²) S(1 –e) + log (e²) S(e) + 2(T(1- e) – T(e)). r² – 1 (Hint: Use the formula for the geometric sum on . Later use integration by parts.) (c) Show that 1-« log (r²) dr = r2 -1 1 22 72k + 1)² lim k-0 (2k + 1)² ° (Hint: Use Abel's theorem to get uniform convergence of T on [0, 1].)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Question 4. Define the power series
p2k+1
S(r):= L 2k + 1
2k+1
and T(x):= E
(2k + 1)2
k=0
km0
(a)
Show that
lim log(1 - a)S(1- a) = 0 and
lim log(r)S(x) = 0.
(Hint: For the first one, recall from $3.4.1 of lecture notes that |r| < }
|log(1 + x)| <x. Use L'Hôpital's rule (more than once if necessary). For the
second one show that r e (0, 1) = S(r) s.)
(b)
Given e e (0,), show that
- log (2²) dz = - log (1- e)?) S(1- e) + log (e?) S(e) + 2(T(1- €) – T(e)).
x² – 1
(Hint: Use the formula for the geometric sum on , Later use integration by
parts.)
(c)
Show that
1-e log (r²)
00
1
lim
E-0+
dr = 22 2k + 1)²
r2 - 1
k=0
(Hint: Use Abel's theorem to get uniform convergence of T on [0, 1].)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc324b15-e30b-4252-860c-612508f5a49d%2F06e88fae-c999-4205-8c9b-ed6b45fcb99e%2Frnerdcl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 4. Define the power series
p2k+1
S(r):= L 2k + 1
2k+1
and T(x):= E
(2k + 1)2
k=0
km0
(a)
Show that
lim log(1 - a)S(1- a) = 0 and
lim log(r)S(x) = 0.
(Hint: For the first one, recall from $3.4.1 of lecture notes that |r| < }
|log(1 + x)| <x. Use L'Hôpital's rule (more than once if necessary). For the
second one show that r e (0, 1) = S(r) s.)
(b)
Given e e (0,), show that
- log (2²) dz = - log (1- e)?) S(1- e) + log (e?) S(e) + 2(T(1- €) – T(e)).
x² – 1
(Hint: Use the formula for the geometric sum on , Later use integration by
parts.)
(c)
Show that
1-e log (r²)
00
1
lim
E-0+
dr = 22 2k + 1)²
r2 - 1
k=0
(Hint: Use Abel's theorem to get uniform convergence of T on [0, 1].)
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