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Problem 31. Without using Taylor's Theorem, represent the following func- tions as power series erpanded about 0 (i.e., in the form Eno an"). (a) In (1 – a²)

Problem 31. Without using Taylor's Theorem, represent the following func- tions as power series erpanded about 0 (i.e., in the form Eno an"). (a) In (1 – a²)

Advanced Engineering Mathematics
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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5:03 < RealAnalysis-ISBN-fix... CALCULUS IN THE 17TH AND 18TH CENTURIES 48 Problem 31. Without using Taylor's Theorem, represent the following func- tions as power series erpanded about 0 (i.e., in the form E-o anr" ). (а) In (1— 2?) (b) (c) arctan (r) (d) In (2 + x) [Hint: 2+x = 2 (1+ 5)/ Problem 32. Let a be a positive real number. Find a power series for a expanded about 0. [Hint: a² = eln (a*)/ Problem 33. Represent the function sin x as a power series erpanded about a (i.e., in the form Eo an (x – a)"). [Hint: sin x = sin (a + x – a)./ Problem 34. Without using Taylor's Theorem, represent the following func- tions as a power series expanded about a for the given value of a (i.e., in the form E-o an (x – a)"). (a) In x, a = 1 (b) е*, а %3D 3 (c) r3 + 2x2 + 3 , a = 1 (d) !, a = 5 Problem 35. Evaluate the following integrals as series. (a) e* dr 1 (b) (c) Next Dashboard Calendar To Do Notifications Inbox 因
Transcribed Image Text:5:03 < RealAnalysis-ISBN-fix... CALCULUS IN THE 17TH AND 18TH CENTURIES 48 Problem 31. Without using Taylor's Theorem, represent the following func- tions as power series erpanded about 0 (i.e., in the form E-o anr" ). (а) In (1— 2?) (b) (c) arctan (r) (d) In (2 + x) [Hint: 2+x = 2 (1+ 5)/ Problem 32. Let a be a positive real number. Find a power series for a expanded about 0. [Hint: a² = eln (a*)/ Problem 33. Represent the function sin x as a power series erpanded about a (i.e., in the form Eo an (x – a)"). [Hint: sin x = sin (a + x – a)./ Problem 34. Without using Taylor's Theorem, represent the following func- tions as a power series expanded about a for the given value of a (i.e., in the form E-o an (x – a)"). (a) In x, a = 1 (b) е*, а %3D 3 (c) r3 + 2x2 + 3 , a = 1 (d) !, a = 5 Problem 35. Evaluate the following integrals as series. (a) e* dr 1 (b) (c) Next Dashboard Calendar To Do Notifications Inbox 因
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