Using Weiestrass M-test, show that the following series converges uniformly.
To show that the series \(\sum_{n=1}^{\infty }n^{3}\: x^{n}\) converges uniformly on the interval \(\left[-\frac{1}{3}, \frac{1}{3}\right]\) using the Weierstrass M-test, we need to find a sequence of bounds \(M_n\) such that: 1. Each term \( |n^3 x^n| \leq M_n \). 2. The series \(\sum_{n=1}^{\infty} M_n\) converges. ___ ______ ___ _____ _____ ________ ___ ___ ____.
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Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:
i) Every infinite set is an open set.
ii) The negation of p∧ ~ q is p → q.
iii) −1is a limit point of the interval ]−2 ,1],
iv) The necessary condition for a function to be integrable is that it is continuous.
v) The function defined by
is differentiable at
Show that is conditionally convergent.
Let be any sequence. Show that
iff for every ε > 0, there exists some
such that n ≥ N implies
Show that the series does not converge uniformly on the interval
Let be a differentiable function on
and
Show that, if
and
then
must have a local maximum at
.
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
Let
be a function defined by
where
Find the values of m and n such that the Rolle’s Theorem holds for the function