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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Find the following limit
Write the inequality in the modulus form.
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
Use Cauchy's mean value theorem to prove that:
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.
Determine the local minimum and local maximum values of the function f defined by .
Show that the equation has a real root other than