Question 178
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function having finite derivative at $0$, and
$$
I(h) = \int_{-h}^{h} f(x) \, dx, \quad h \in [0, 1].
$$
Prove that
a) There exists $M \gt 0$, such that $|I(h) - 2f(0)h| \leq Mh^2$, for any $h \in [0, 1]$.
b) The sequence $(a_n)_{n \geq 1}$, defined by $a_n = \sum_{k=1}^{n} \sqrt{k} |I(1/k)|$, is convergent if and only if $f(0) = 0$.
$$
I(h) = \int_{-h}^{h} f(x) \, dx, \quad h \in [0, 1].
$$
Prove that
a) There exists $M \gt 0$, such that $|I(h) - 2f(0)h| \leq Mh^2$, for any $h \in [0, 1]$.
b) The sequence $(a_n)_{n \geq 1}$, defined by $a_n = \sum_{k=1}^{n} \sqrt{k} |I(1/k)|$, is convergent if and only if $f(0) = 0$.