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Check whether the sequence (an), where


an = 1/ (n+1) + 1/(n+2) +....+1/(2n) is convergent or not




Show that if 5/3< 2x< 11/3, then x∈{y∈R such that |y- 4/3| < 1/2}.

Assume that (M, d) is a compact metric space. Show that if f : (M, d) → (Y, ˜d) is continuous and bijective,


then f is a homeomorphism.

Let f:[ 0,π/2] → [-1,1] be a function defined by f(x)= cos 2x . Verify that f satisfies the condition of the inverse function theorem. Hence, what can you conclude about the continuity of f^-1?

Prove that the sequence (fn(x)), where fn(x)= nx/(1+ nx^2) is not uniformly convergent in [-2,2]

Check whether or not the function f, defined on R by


f(x) = { 3x^2sin(1/2x), when x≠0


{ 0 ,when x=0


is derivable on R. If it is, is f' continuous at x=0? If f is not derivable , then define a derivable function on R

Let the function , defined by


f(x)= 1/x+3 ; x∈ [3,∞[



Check whether f is uniformly continuous or not on the interval of definition

prove or disprove that the complement of integers z in R is an open set


Evaluate


lim 2r Σ r=1 [2n^2/(n+r)^3]


n→∞

Check whether or not the sequence (n- 1/n) is convergent

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