Real Analysis Answers

Verify bounded convergence theorem for the sequence of function fn(x)= 1/(1+x/n)^n,x varies from 0 to 1

Verify bounded convergence theorem for sequence of function
fn(x)=1/(1+x/n)^n,0<=x<=1

If the Fourier series of a function f is Cesaro summable, then show that it is Abel summable?

Find sup A, inf A, B sup, inf B and justify your answer. Determines whether the borders are rational and decide how the borders for A and B lies respectively for A and B.

5. (a) Define a compact set. Check whether the
set of integers is compact or not. 2
(b) If a + b + c = — 4 and 4a + 2b + c = 6, then
show that both the roots of the quadratic
equation axe + bx + c = 0 are real.
(c) Using Taylor's Theorem, prove that
, x2 x4
+ V XE R.
2! 4!
6. (a) State the second mean value theorem of
integrability. Verify it for the functions
f and g defined on [1, 2] by fix) = 3x and
g(x) = 5x.
(b) Test the following series for absolute and
conditional convergence : 5
00
(i) E E1)n
3n+1
n=1
E00 sin nx
n3
n=1
4
4
5
MT E-09 4
7. (a) Prove that there is no rational number
whose square is 6. 3
(b) Check whether the following functions are
continuous or not at x = 0. Also, find the
nature of discontinuity at that point, if it
exists. 4
- 2 x
(i) f(x) =
1
1
X
, x 0
, x=0
X
2 + -1 x<_0 ,
(ii) f(x) =
3
- (X3 -F ) , x>0 3
(c) Examine the function f given by
ME) = (x - 8)3 (x + 3), x E R
for extreme values. .

co 7n
(3n + 1) !
n=1
00
1
n=1
(i)
3. (a) Show that the sequence (fn) where
fn(x) = x e [1, .[ is uniformly
1 +2ne
convergent in [1, .4.
(b) Check whether the following sequences (s n)
are Cauchy, where
(i) sn =1+2+3+...+n
4n3 + 3n (ii)
sn 3n + n2
(c) Check whether the function f(xx)) = cos —1 is
x
uniformly continuous on the interval JO, 1[.
Is it continuous on the same interval ?
Justify.
4. (a) Show that the union of two open sets is an
open set.
(b) Verify Inverse Function Theorem for finding
the derivative at a point yo of the domain of
the inverse function of the function
f(x) = cos x, x E [0, 7C] . Hence, find the
derivative at yo.
(c) Test for convergence the following series : 4
3
4
3
3