Real Analysis Answers

Check whether the function, f , defined below, is uniformly continuous or not:

f (x) =x^1/2 , x belong to [1,2].

Use Lagrange’s mean value theorem to prove that

x-x^2/2<log(1+x)<x-x^2/2(1+x) if x>0

Test the absolute and conditional convergence of the series summation n=1 to infinity (-1)^n/2n+7

Check whether the sequence, {Sn } WHERE

,Sn =1/1!+1/3!+1/5!+...+1/(2n-1)!



Is convergent or not.

the following series 1-1/2^2-1/3^2+1/4^2-1/5^2-1/6^2+... is convergent or not?

Let a function f :R to R be defined by

f(x)={2if x belong to Q,4 if x does not belong to Q


CHECK WHETHER f is continuous on B

Ï

Î

=

Q

Q

4, if x

2, if x

f x

Check whether f is continuous on B.

prove the following series 1-1/2^2-1/3^2+1/4^2-1/5^2-1/6^2+... is convergent

Show that if X and Y are sequences such that X and X+Y are convergent, then Y is convergent.

a few questions:

1.Give a neighbourhood of a point 7.3

2. Is [0,1] an open set?

Is V=(1,2) a neighbourhood of the point 1.6 ? what epsilon neighbourhood would work ? Five value of epsilon