Check whether the series is uniformly convergent or not, wher
To check the uniform convergence of the given series \[ f(x) = 2x + 7 \sum_{n=1}^{\infty} \frac{n^2 x^5}{n^4 + x^3}, \quad x \in [0, \alpha], \] where \( \alpha \in \mathbb{R}^+ \), we need to examine if the series converges uniformly on the interval \( [0, \alpha] \). ### Step 1: _______ ___ __________ ________ _______.
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Let be any sequence. Show that
iff for every ε > 0, there exists some
such that n ≥ N implies
Prove that a strictly decreasing function is always one-one
d) Show that is a Cauchy sequence.
Use Cauchy’s Mean Value Theorem to prove that:
Let be a differentiable function on
and
Show that, if
and
then
must have a local maximum at
.
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
Give an example of a divergent sequence which has two convergent subsequences. Justify your claim.