Test the following series for convergence.
(i)
(ii)
Let's solve and analyze the convergence of these series. ### (i) Series: \(\sum_{n=1}^{\infty } n \cdot x^{n-1}, \quad x > 0\) We recognize this as a geometric series with the term \(n \cdot x^{n-1}\). To test for convergence, we first check whether the series is summable using the ratio test or by recognizing it as a known series type. For the series \(\sum_{n=1}^{\infty } n \cdot x^{n-1}\), the general term is \(a_n = n \cdot x^{n-1}\). We can apply the ratio test to find ____ ___ ________ ________ _____ ____ __________.
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Suppose that is continuous on
and differentiable on
and that f )0( = ,0 f )1( = ,1 f )2( = .1 (i)
Show that there exists such that
Show that there exists such that
(iii) Show that there exists such that .
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
Check whether the series is uniformly convergent or not, wher
Give an example for each of the following.
i) A set in with a unique limit point.
ii) A set in whose all points except the one are its limit points.
iii) A set having no limit point.
iv) A set S with
v) A bijection from to
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
Show that the series does not converge uniformly on the interval
Prove that a strictly decreasing function is always one-one