Real Analysis Answers

Evaluate

lim 2r Σ r=1 [2n^2/(n+r)^3]

n→∞

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}.

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

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

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

Check whether the interval [7,10[ and ]3,6] are equivalent or not

If (an) is convergent then "\\sum_{i=1}^{\\infty} a_n" is also convergent. True or false with the full explanation.