Answer to Question #67005 in Real Analysis for koel

Question

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