Using Weiestrass M-test, show that the following series converges uniformly.
___ ____ ___ _______ ______ ___ __________ ____ ________ __________.
_______ _____ _________ _______ __________ _______ ____ _________ _______ ________.
___ __________ ______ _____ _______ ________.
______ __________ _________ _________ ________ ___ ____ _______ _______ ______.
_______ _______ _______ __________ ________ ___.
_________ ___ ______ _____ ________ _____ _______ ______ ___ ________ ______.
_______ ____ ________ __________ ____ __________ _______ _________.
______ ________ _____ ______ __________ _____ _______ _____ ______.
______ ____ _________ ____ __________ ___ ________ ________ ________.
_______ __________ ___ _________ ____ ____ ____ ________ ___.
______ __________ ___ ______ ________ _______ _________ __________ ______ _______ __________.
_____ ___ _______ ______ _______.
______ __________ ______ ____ _________ _____ __________ ________ _____ _____.
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Use Cauchy's mean value theorem to prove that:
a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
Write the inequality in the modulus form.
Determine the local minimum and local maximum values of the function f defined by .
Show that is an algebraic number.
Check whether the set of integers is countable or not.
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.