a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
(i) We are asked to test the convergence of the series [ \sum_{n=1}^{\infty} n x^{,n-1}, \quad x > 0. ] We can use the ratio test. Let (a_n = n x^{,n-1}). Consider [ \frac{a_{n+1}}{a_n} = \frac{(n+1)x^n}{n x^{,n-1}} = \frac{n+1}{n} \cdot x = \left(1 + \frac{1}{n}\right)x. ] Take the limit as (n \to \infty): [ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} _____ __________ __________ ________ ______ ________ _________ ____ _____ ________ _____ __________.
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Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Using Weiestrass M-test, show that the following series converges uniformly.
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.
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.
Find a and b such that exists.
Show that is an algebraic number.