Real Analysis Answers

The function f: [ 2,4] to R , defined by f(x)= 3/x is uniformly continuous on its domain.

True or false with full explanation

Check whether the sequence { an} , where an= 1/(n+1)+ 1/(n+2)+..1/(2n) is convergent or not.

If a function f:[ a,b] to R has finitely many points of discontinuity in [ a,b] , then f is integrable on [ a,b].

True or false with full explanation

Prove that the sum of two convergent sequence is convergent.

1. Show that the function f(x)= | cos 2x| is a periodic function

2. Find the local extreme value of (1/x)^x , if it exists.

∞Σn=1 sin(1/n) is a convergent series.

True or false with full explanation

Show that the set [ -5,3] ∩[ -3,5] is a neighborhood of 2.

Use the principle of mathematical induction to show that

| sin nx| ≤ n| sin x|

for all n∈ N and for all x ∈ R

Show that the equation

2x^3-3x^2+7x-18=0 has a real root which is real and positive