4. Prove that given a continuous function f on [a, b], there exists a sequence (Pn) of polynomialssuch that pn →f uniformly on [a, b], and for each n, pn(a) = f(a) and pn (b) = f(b).Hint: Select a sequence (an) of polynomials such that qnf uniformly on [a, b]. For eachn, let sn be the function on R whose graph is the straight line passing through the points(a, f(a)- qn(a)) and (b, f(b)- qn (b)). Set pn = 9n + Sn.

Answered: Image /qna-images/answer/2220ee5b-8c3e-4701-aab9-d0a93776ea21.jpg

Prove that given a continuous function f on [a, b], there exists a sequence (Pn) of polynomials such that pn →f uniformly on [a, b], and for each n, pn(a) = f(a) and pn(b) = f(b). Hint: Select a sequence (gn) of polynomials such that qnf uniformly on [a, b]. For each n, let sn be the function on R whose graph is the straight line passing through the points (a, f(a)- qn(a)) and (b, f(b) — qn (b)). Set pn = qn + Sn.

Prove that given a continuous function f on [a, b], there exists a sequence (Pn) of polynomials such that pn →f uniformly on [a, b], and for each n, pn(a) = f(a) and pn(b) = f(b). Hint: Select a sequence (gn) of polynomials such that qnf uniformly on [a, b]. For each n, let sn be the function on R whose graph is the straight line passing through the points (a, f(a)- qn(a)) and (b, f(b) — qn (b)). Set pn = qn + Sn.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 74E

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