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} = x. ] * If (x < 1), the ratio (< 1), series converges. * ____ ___ _________ ________ ________ ______ ____ ____.
______ _____ _____ ______ ____ ____ _____ ________ ___ __________ ______.
_______ _______ __________ ________ _________ __________ ___ ______ _________ _________ ____ _____.
_________ ______ _______ ___ ____ ________ ________ _________ ___.
___ ____ ________ ____ ____ ___ _________ ____ ________ _________ ___.
_____ __________ ________ ______ __________ __________.
_______ ___ ______ ____ __________ ___ ____ _____ ______.
_________ ________ ____ ______ ________ ___ ___ __________ __________.
__________ _____ ____ _________ ________ _____ __________ ______ ___ ______.
_______ _______ _____ ______ ___ __________ ___ _________ _____ __________ ______ _____.
___ ____ _______ ______ ___ ____ _____ _____ ________ _______ _________.
_______ _____ ____ _________ ______ ____ ___ ___ ____ _______ ____.
____ _____ _____ _______ _____.
_____ ____ ___ ___ ______ __________ ______ ________.
______ ________ ________ ______ __________ _________ __________ _______ _____ ______.
__________ ____ _____ __________ ___ ______ _________ ________.
_____ _________ ________ _________ ___ ______ _____.
____ ________ ____ _______ _________ ________ ____ __________ _________ ______.
___ ______ _______ ______ _____ _______ _____.
__________ ______ _______ __________ __________ ________ _____ __________.
____ _____ ______ ______ ______ ____ ________.
________ _____ ___ ________ ____.
_______ ____ __________ ________ ________ ________ ____ ____ _______ _______.
________ _____ ________ ________ ______ ________ ____ ___.
________ ____ ____ ___ _______ ______.
_____.
Get Full Answer on WhatsApp
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Show that is an algebraic number.
Prove that a strictly decreasing function is always one-one.
Check whether the set of integers is countable or not.
a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
">
Also check whether the function f is derivable at x = 1.
Apply Bonnet Mean Value Theorem for integrals to show 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.