Sequences and Series of Functions — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

ro) show that (c) let f(X)=Z:=oanxn be a power series with finite positive radius of convergence R. Prove that if the series converge at x = R, then f is continuous Semester at x = R. If the. series converges at x = - R, then f is continuous at x = -R : Sequences and Series ofFhnctions@SC) TSne : 8 Hours Ma}ckmum Men:ks : 90 Letfbe acontinuous function on [a,b]. Showthat . there exists a sequence (pj of pcilynomials such that pn + f uniformly on [a,b] and pn(a) = f(a) for If x.a 0 a.nd x € N , show that Hence show that 1. (a) State and prove Cauchy's criterion for uniform in(1+x)=x-¥+i-..........+(-1)n¥+toxgdt •. (c) Show that if x > 0 then Use 'this inequality to establish a lower bound for 7t ` convergence for Sequence of functions® (b) Show that the sequence (n2x2e-nx) converges uniformly on the interval [1, oo ), but that it does not converge uniformly in [0, co ). (c) (i) Show that if the sequences`(fn) , (gn)iconverge uniformly on the set A to f and g, respectively, then the sequence (fn+gn) converges uniformly to. f + g . (ii) Discuss the uniform convergence of the tds* sequence of functions (¥)onR 2. (a) Let (f) be a sequence of continuous functions →R. Prove that f is continuous on A. (b) Let f(x) - I+ nx for XEA [0,00). Show that each ¿ is bounded on A, but the pointwise limit f of the sequence (f.) is not bounded on A. Does the sequence (f) converge uniformly to f on A? (c) (i) Let f, (x)= -, for xE [0,1]. show that the sequence (f.) of differentiable functions onverges uniformly to a differentiable unction f on [0,1], and that the sequenc (f'.) converges on [0,1], but that g(1) + f'(1). (i) 1r aso, show that lim i Sini ax -0 3. (a) State and prove Weierstrass M-test for uniform convergence of the series of functions. Hence, show that. Eb, sinnx is a uniformly convergent series given that Eb, is an absolutely convergent series. (b) Let f be a real valued function on la, b] that has derivative f; on la, bl for each neN. Suppose that the series' Ef, converges for at least one point of la,b) and the series of derivatives If converges uniformly on la,b]. Show that there exists a real valued function f on [a, b] such that Ef, converges uniformly to f on [a,b]. Also, that f has a derivative on (a,b] and f'= Ef (c) Discuss the pointwise convergence of Enly +T*≥0. Also, show that the given series is uniformly convergent on (2, 0) but not uniformly convergent on (1,0). 4. (a) Frove that Es) converges for all xeR. .. Also, show that the series converges uniformly on I-a, al, a>0 but convergence is not uniform on R. (b) Let (f) be a sequence of integrable functions on la,bl and suppose that Enf converges uniformly to f on [a,b]. Show that f is integrable (c) Prove that In? is convergent for all x*0. Also, show that the convergence is not uniform for |x|<1 but is uniform for |x|≥1. 5. (a) Define power series and its radius of convergence. Evaluate the radius of convergence for the series