4. Given a ER and n e N, the binomial coefficient (@) is defined by the formula a(a – 1)(a – 2) · · (a – n+ 1) п! (a) Let a e R and let f(x) = (1+x)ª where r > 0. Prove that f(m) (x) (1+x)ª-" for all n e N. a-n n! Hence derive a formula for the Maclaurin series of (1 + x)ª. (b) Compute the Maclaurin polynomial of (1+ x)-1/2 of order 4. (c) Compute the radius of convergence of the Maclaurin series of (1+ x)-1/2 using the ratio test.
4. Given a ER and n e N, the binomial coefficient (@) is defined by the formula a(a – 1)(a – 2) · · (a – n+ 1) п! (a) Let a e R and let f(x) = (1+x)ª where r > 0. Prove that f(m) (x) (1+x)ª-" for all n e N. a-n n! Hence derive a formula for the Maclaurin series of (1 + x)ª. (b) Compute the Maclaurin polynomial of (1+ x)-1/2 of order 4. (c) Compute the radius of convergence of the Maclaurin series of (1+ x)-1/2 using the ratio test.
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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Transcribed Image Text:4. Given a ER and n e N, the binomial coefficient (@) is defined by the formula
a(a – 1)(a – 2) · · (a – n+ 1)
п!
(a) Let a e R and let f(x) = (1+x)ª where x > 0. Prove that
f(m) (x)
(1+ x)ª¬n for all n e N.
a-n
n!
Hence derive a formula for the Maclaurin series of (1 + x)ª.
(b) Compute the Maclaurin polynomial of (1+ x)-1/2 of order 4.
(c) Compute the radius of convergence of the Maclaurin series of (1+x)-1/2 using the ratio test.
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