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Real Analysis
2.
(c)
(d)
(e)
(a)
The greatest integer
R is derivable in
lim cot x does
function
the interval
not exist.
(7c
of Mathematical
is convergent.
[-1
2 4
[x] defined on
Induction,
It It
-> - X --
2 2
The series
nZ
Using the Principle
show that
1 + 3 + 5 + 7 + + (2n - 1) = n2, V n E N.
(b) Show that the function f : [0, 1] -4 R
defined by
11 when x is rational
f(x) =
is not Riemann integrable. 4
(c) Show that the function
f(x) = I x - 5 I + x2 + 3x + 10
is continuous but is not differentiable at the
point x = 5
(c)
(d)
(e)
(a)
The greatest integer
R is derivable in
lim cot x does
function
the interval
not exist.
(7c
of Mathematical
is convergent.
[-1
2 4
[x] defined on
Induction,
It It
-> - X --
2 2
The series
nZ
Using the Principle
show that
1 + 3 + 5 + 7 + + (2n - 1) = n2, V n E N.
(b) Show that the function f : [0, 1] -4 R
defined by
11 when x is rational
f(x) =
is not Riemann integrable. 4
(c) Show that the function
f(x) = I x - 5 I + x2 + 3x + 10
is continuous but is not differentiable at the
point x = 5
Real Analysis
(a) Find the sum of the series
VT).
+ + + ,
./ v (n + 3)3 11(n + 6)3 V[n 3 (n — 1)]3
3
(b) By showing that the remainder after n-terms
tends to zero, find Maclaurin's series
expansion of sin 2x. 5
(c) Prove that the function f defined on [0, 1] by
f(x) = (— 1)n+1for 1
< x < ,
n + 1
n = 1, 2, 3, ... is integrable on [0, 1].
VT).
+ + + ,
./ v (n + 3)3 11(n + 6)3 V[n 3 (n — 1)]3
3
(b) By showing that the remainder after n-terms
tends to zero, find Maclaurin's series
expansion of sin 2x. 5
(c) Prove that the function f defined on [0, 1] by
f(x) = (— 1)n+1for 1
< x < ,
n + 1
n = 1, 2, 3, ... is integrable on [0, 1].
Real Analysis
1. Which of the following statements are true or
false ? Give reasons for your answer. 10
(a) The function f(x) = 3x2+ 5 I x I is
differentiable at x = — 2.
(b) The singleton set {x} for any x E B is an open
set.
(c) Every bounded sequence is convergent.
(d) Every integrable function is monotonic.
(e) The function fix) = cos x is uniformly
continuous on [0, —E 2l
false ? Give reasons for your answer. 10
(a) The function f(x) = 3x2+ 5 I x I is
differentiable at x = — 2.
(b) The singleton set {x} for any x E B is an open
set.
(c) Every bounded sequence is convergent.
(d) Every integrable function is monotonic.
(e) The function fix) = cos x is uniformly
continuous on [0, —E 2l
Real Analysis
Q. Define the following terms
Series
Conditionally convergent series
Completeness property
Series
Conditionally convergent series
Completeness property
Real Analysis
Compute the Riemann integral of the function f (x) = x on the interval [−1, 1].
Real Analysis
Verify Inverse function theorem for finding the derivative at a point y of the 0
domain of the inverse function of the function f (x) = cosx, x ∈[0,π].Hence, find
the derivative of the inverse function aty
domain of the inverse function of the function f (x) = cosx, x ∈[0,π].Hence, find
the derivative of the inverse function aty
Real Analysis
Verify the second mean value theorem of integrability for the functions f and g
defined on [1,2] by f (x) = 3x and g(x) = 5x.
defined on [1,2] by f (x) = 3x and g(x) = 5x.
Real Analysis
Q: show that if zn=(an+bn)1/n where 0<a<b, then lim(zn)=b.
Real Analysis
Q: Use the sequeeze theorem to determine the limits of the following:
(a)n1/n^2, (b)〖 ((n!)〗^(1/n^2 ))
(a)n1/n^2, (b)〖 ((n!)〗^(1/n^2 ))
Real Analysis
Q: If a>0, b>0, show that lim(√((n+a)(n+b))-n)=(a+b)/2
