Elementary Mathematical Analysis — DU B.Sc. (Prog) Physical Sciences courses pertaining to Mathematical Sciences Previous Year Question Papers

 n is pointwise convergent but not uniformly convergent on [0, 1]. (2.5,5) (b) State Weierstrass M-Test and show that the series: n is uniformly convergent on  ,   . (2.5,5) (c) Find the radius of convergence of the power series: 7.5 6. (a) Show that the sequence n f   , where x f x , n N 1 n x is uniformly convergent on [0,1]. (b) Show that the series. x n e is uniformly convergent on  ,   . 7.5 (c) Find the interval of convergence of the power series. 7.5 1. (a) In each of the following, a function f(x) and a number x0 are given. Use the sequential criterion to prove that lim f x x x  does not exist. (b) Use the sequential criterion to find the limit of the function f(x) at x0= 2 if it exist where f x 0, if x is irrational and show that the function f(x)=sin(x) is uniformly continuous on . (3.5,4) (c) Using the sequential criterion, show that if lim f x L  , where x0 is a cluster point of domain of both functions f and g. 2. (a) Consider the function T :    , defined by n n where m ,n with m and n have no common prime factors Show that T(x) is continuous at every irrational number, but is discontinuous at every rational number. (7.5) (b) Let f: [a,b] [a,b] be a continuous function. Prove that there exists c[a,b] such that f(c) = c 7.5 (c) Prove that x x2 I for some   x 0,1  . 7.5 3. (a) Define upper and lower (Darboux's) sums of f corresponding to the partition P of [a, b] for the function f(x) = x²+3x on [0, 2], find upper and lower sums for partition (b) If Q is a refinement of a partition P, then prove that (c) If f is monotone on [a, b], then prove that f is integrable on [a, b]. 7.5 4. (a) Use sequential criterion for integrability to show that function f(x) = x²–2x is integrable on [2, 5] and also find (b) If f is integrable on [a, b], then prove that |f| is also integrable, further show that a a where, M is upper bound for |f| on [a,b]. Find a function f : [0,1]   such that f2 is integrable on [0,1] but f is not. (c) State and prove the Fundamental Theorem of Calculus (first form). 5. (a) Define pointwise convergent sequence and show that the sequence, n f   where: