Real Analysis Answers

If f is monotonic on (a,b) then show that f is of bounded variation on(a,b)

{F} check,whether the collection G,given by:

G'={]1/n+2,1/n[:n€N}

is an open cover of ]0,1[

{F} test the series : infinity sigma n=1 (-1) ^n-1 sin nx/n√n for absolute and conditional convergence

<e> Examine the f:R->R defined by

f(x)={1/6(x+1)^3 x is not equal to 0

5/6 x=0}

for continuity on R .If it is not continuous at any of R find the nature of discontinuity there

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

True or false with full explanation

Does the sequence 3+ (-1)^n converge to 2. Justify

Does the sequence 3+ (-1)^n converge to 2. Justify