For $0<x<1$, the Abel transform of a series is $(1-x{)}^2{\sum }_{n\ge 0}(n+1){\sigma }_n{x}^n$, where ${\sigma }_n$ are the Cesaro means of its partial sums

Given: The coefficients, partial sums, and Cesaro means in the statement.

The canonical natural $\iota (n+1)$ is positive. Thus, putting $T_n:={\sum }_{k=0}^n{S}_k$, the definition of ${\sigma }_n$ gives $T_n=\iota (n+1){\sigma }_n$. Also $S_n=T_n-T_{n-1}$ with $T_{-1}:=0$; putting $S_{-1}:=0$ gives $a_n=S_n-S_{n-1}$ for every $n\ge 0$ (Abel summability by ${\lim }_{x\uparrow 1}\sum a_n{x}^n$ and Cesaro summability by the Cesaro means of the partial sums, Finite sums and finite products, by recursion, The canonical natural $\iota (n)=n\cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

For $0<x<1$, the geometric series is absolutely convergent because its terms are nonnegative. Its Cauchy product with itself has coefficient ${\sum }_{k=0}^n{x}^k{x}^{n-k}=\iota (n+1)x^n$ and sum $1\mathrm{/}(1-x{)}^2$ (For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges, Monotonicity of $x\mapsto x^n$ and of $n\mapsto a^n$, Basic properties of the absolute value, The Cauchy product of two series: $c_n={\sum }_{k=0}^n{a}_k{b}_{n-k}$, If $\sum a_k$ and $\sum b_k$ both converge absolutely then their Cauchy product converges absolutely, with sum $AB$, Finite sums and finite products, by recursion, The canonical natural $\iota (n)=n\cdot 1_F$ of a field).

Convergent real series may be added, subtracted and scaled term by term (Convergent series add and scale termwise).

A nonnegative series dominated termwise by a convergent nonnegative series converges (If $0\le a_k\le b_k$ eventually, convergence of $\sum b_k$ gives convergence of $\sum a_k$, and divergence of $\sum a_k$ gives divergence of $\sum b_k$).