Using the principle of mathematical induction, show that
We are asked to prove by mathematical induction that [ 1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{1}{3} n(4n^2 - 1), \quad \forall n \in \mathbb{N}. ] Step 1: Base Case (n = 1) Left-hand side (LHS): (1^2 = 1) Right-hand side (RHS): (\frac{1}{3} \cdot 1 \cdot (4 \cdot 1^2 - 1) = \frac{1}{3} \cdot (4 - 1) = \frac{3}{3} = _________ ______ ___ __________ ______ ____ ____ _______ ________ _______ ____ ________.
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Show that the equation has a real root other than
Using Weiestrass M-test, show that the following series converges uniformly.
Use Cauchy's mean value theorem to prove that:
Using the principle of mathematical induction, show that
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.
a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
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Also check whether the function f is derivable at x = 1.