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].)
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 70E
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