Real Analysis Answers

R is an ordered eld which properly contains Q as an ordered subfield.

(a) Does R have elements 0 and 1 such that x+0 = x for all x 2 R and x
1 = x for

all x 2 R.

(b) Prove that 0 = 0 and 1 = 1 where 0 and 1 are the integers that we know very

well.

If 𝑓 is a continuous function on [0,1], show that lim _n->infinity ∫ ¹₀(𝑛𝑓(𝑥) )/(1+𝑢²𝑥 ² ) 𝑑x = 𝜋/ 2 𝑓(0). 

Check the convergence of the sequence defined by 𝑢_𝑛+1 =1/2 (𝑢_𝑛 + 𝑎 /𝑢_𝑛 ) , 𝑎 > 0.Note that this is the sequence associated with finding the square root of a number 𝑎 > 0 by the Newton’s method.

Use the Taylor series to find the values of ln 0.4 accurate to 10^-3 . Use the integral remainder.

Use the Taylor series to find the values of ln 0.4 accurate to 10^-3 . Use the integral remainder.

Show using an example 𝑓 and 𝑔 are not integrable on [𝑎, 𝑏], but 𝑓𝑔 may be integrable on [𝑎, 𝑏].

Use the Taylor series to find the values of ln 1.4 accurate to 10^-3 . Use the integral remainder.