Let $R$ be a commutative ring, let $(a_n)_{n\geq 0}$ and $(b_n)_{n\geq 0}$ be sequences with values in $R$, and let $\lambda\in R$. For a formal [power series](/page/Power%20Series) $F(X)=\sum_{n=0}^{\infty} r_n X^n\in R[[X]]$, write $[X^n]F(X)=r_n$ for the coefficient of $X^n$; equivalently, $F(X)$ is the ordinary [generating function](/page/Generating%20Function) of the sequence $(r_n)_{n\geq 0}$. Define formal power series $A(X),B(X)\in R[[X]]$ by

\begin{align*} A(X)=\sum_{n=0}^{\infty}a_nX^n \end{align*}

and

\begin{align*} B(X)=\sum_{n=0}^{\infty}b_nX^n. \end{align*}

Then $A(X)+B(X)$ is the ordinary generating function of the sequence $(a_n+b_n)_{n\geq 0}$, and $\lambda A(X)$ is the ordinary generating function of the sequence $(\lambda a_n)_{n\geq 0}$.