Real Analysis Answers

Let Pun be a series of arbitrary terms. For all n ∈ N, let pn =

1

2

(un + |un|) and qn =

1

2

(un − |un|). Show

that

(a) If Pun is absolutely convergent, then both Ppn and Pqn are convergent.

(b) If Pun is conditionally convergent, then both Ppn and Pqn are divergent.

Give an example of a function in L₃[1 2] that is not in C [1 2]

Consider the sequence {xn}, which is defined by

x1 = 1, xn+1 = xn +

1

x1 + x2 + · · · + xn

, n ∈ N.

Does {xn} converge? Justify your answer.

(2)

? ? ? ? ?

Using the sequential definition of 
the continuity, prove that the funct-ion, f defined by
f(x)= { 3 , if x is irrational
      { -3 , if x is rational
 is discontinuous at each real numbers.

State Bonnet's mean value theorem for integrals.

Apply it to show that:

5

| ∫ (cos x/x)dx | ≤ 2/3

3

Examine the continuity of the function:

f: [ 1,3]→R defined by :

f(x)= [x]/3x-1 where [x] denotes the greatest integer function

Show that on the curve, y= 3x^2-7x+6, the chord joining the points whose abscissa are x=1 and x=2, parallel to the tangent at the point whose abscissa is x=3/2

Check whether the function f given by:

f(x)= (x-4)^2(x+1)^4 has local maxima and loca

minima

For each of the following statements give the converse, the contrapositive and the negation of the statement.

(i) I take plastic bags when I go shopping.

(ii) x∈Bimpliesx∈/Xorx∈/Y.

(iii) Ifx∈A∩Bthenx∈Aand x∈B.

Let S = {a1, a2, a3, ...an}be a set of test scores. Prove using the the indirect method of proof

that if the average of this set of test scores is greater than 90, then at least one of the scores is greater than 90.