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Real Analysis
Find the upper and lower integrals of the function f defined by
f(x)= (7/2)-2x, ∀ x∈[1,3]
Is f integrable over the interval [1,3]? Justify.
f(x)= (7/2)-2x, ∀ x∈[1,3]
Is f integrable over the interval [1,3]? Justify.
Real Analysis
Show that the sequence (f) where f(x)= x/(1+2nx^2), x∈ [1,∞[ is uniformaly convergent in [1,∞[
Real Analysis
Check whether the function f (x) = [x]+ e^x is integrable in [0,3].
Real Analysis
Evaluate the limit as n → ∞ of the sum
1/n [sin (π/n) +sin (2π)/n +.....+ sin (2nπ)/n]
1/n [sin (π/n) +sin (2π)/n +.....+ sin (2nπ)/n]
Real Analysis
Show that the function f :[0, 1] → R defined by
f(x)= { 1, when x is rational ; 2 when x is irrational
is not Riemann integrable.
f(x)= { 1, when x is rational ; 2 when x is irrational
is not Riemann integrable.
Real Analysis
Show that the function f defined on R by
f(x)={3x^(2) cos(1)/(2x), when x≠0 ; 0, when x=0
is derivable on R but f ′ is not continuous at x=0
f(x)={3x^(2) cos(1)/(2x), when x≠0 ; 0, when x=0
is derivable on R but f ′ is not continuous at x=0
Real Analysis
Show that the set B = {x | x^(2) > 2} is non-empty and bounded below. Is it bounded
above? Justify.
above? Justify.
Real Analysis
Evaluate: lim x→∞ {[2x^(2) +3x-2]^(1/2)}-{[2x^(2) -3x+2]^(1/2)}
Real Analysis
Check whether the following sequences{sn} are Cauchy, where
(i) sn = 1+2+3+....+n
(ii) sn = [4n^(3)+3n]/[3n^(3)+n^(2)]
(i) sn = 1+2+3+....+n
(ii) sn = [4n^(3)+3n]/[3n^(3)+n^(2)]
Real Analysis
For the following sequences, find two subsequences which are convergent:
(i) a(subscript n)= n[1+(-1)^n]
(ii) a(subscript n)= sin [(nπ)/3]
(i) a(subscript n)= n[1+(-1)^n]
(ii) a(subscript n)= sin [(nπ)/3]
