Use the Fundamental Theorem of Integral Calculus to evaluate the integral
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Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?
Check whether the set of integers is countable or not.
a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Find the following limit
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.
Prove that a strictly decreasing function is always one-one.