Real Analysis Answers

Find whether the following series are convergent or not

i. ∞Σ n=1 (3n-1)/7^n

Let f be a function defined on R by:

F(x)= { x+5/(x^2-25) , when x≠5

{ 1 when x= 5

Check whether f is uniformly continuous on [-3,3] or not

Show that the function f: [0,1]→ R defined by

F(x) = { 2 , when x is rational

{ 3 , when x is irrational

is not riemann integrable on[0,1]

Give an example of each of the following:

i. a function with a removable discontinuity

ii. a totally discontinuous function

Check whether the sequence {an}, where

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

The function f: R→R defined by f(x)= | x-1|+ | 3-x| is differentiable at x= 4.

True or false with full explanation

The function f: R→R defined by f(x)= | x-1|+ | 3-x| is differentiable at x= 4.

True or false with full explanation

Prove that

lim 1/(x+5)^2 = ∞

x→-5

The function f(x)= [x]- x is not integrable in [ 0,3] , where [ x] denote greatest integer function.

True or false with full explanation