Let {fn}%=1 be a sequence of continous functions on an interval [a, b]. If {fn}-1 converges uniformly to a function f on [a, b], then f is continous on [a, b]. (a) (i) Discuss the convergence of the sequence of functions fn, when fn(x) 1+ æn' where x E [0, 1). 1 (ii) Show that the sequence {fn}, where fn(x) is not uniformly convergent 1+ xn on x € [0, 1]. x2n (b) Let fn(x) = is not uniform. 14 g2n X E [0, 1]. Use the above theorem to show that the convergence (c) Is the converse of the above theorem necessary true? Justify your answer using the sequence fn(x) = nxe¬nª, x E [0, 1].

Let {fn}%=1 be a sequence of continous functions on an interval [a, b]. If {fn}-1 converges uniformly to a function f on [a, b], then f is continous on [a, b]. (a) (i) Discuss the convergence of the sequence of functions fn, when fn(x) 1+ æn' where x E [0, 1). 1 (ii) Show that the sequence {fn}, where fn(x) is not uniformly convergent 1+ xn on x € [0, 1]. x2n (b) Let fn(x) = is not uniform. 14 g2n X E [0, 1]. Use the above theorem to show that the convergence (c) Is the converse of the above theorem necessary true? Justify your answer using the sequence fn(x) = nxe¬nª, x E [0, 1].

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 68E

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