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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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.
Show that , the lagrange's form of remainder in the Maclaurin series expansion of e4x, tends to zero as
. Hence obtain the Maclaurin's infinite expansion for e4x.
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Show that is an algebraic number.
Check whether the set of integers is countable or not.
Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?