Exercise 18, § 3

Let $(a_n)_{n \geq 0}$ be an arbitrary sequence of complex numbers. For each $n \geq 0$ put $s_n^{(0)} = a_n$, and, inductively, for $k \geq 0$, define

a) Prove "Taylor's formula for sequences": for each integer (proceed by induction on $k$).

b) Suppose that there is a number $C$ such that $|n a_n| \leq C$ for all $n$, and that the sequence $(s^{(2)}/n)$ formed by the arithmetic means $(s_0 + \cdots + s_{n-1})/n$ of the partial sums $s_n = a_0 + \cdots + a_{n-1}$ tends to a limit $\sigma$. Show that the series with general term $a_n$ is convergent and has sum $\sigma$ ("Hardy-Littlewood tauberian theorem"). (Write

where $|r_n|$ is bounded above with the aid of the inequality $|n a_n| \leq C$, and $h$ is chosen suitably as a function of $n$.)