$(\lambda-a)^{-1}$ as limits of 'polynomials' For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation}
\sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a)
\end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$. 

It is natural to look at the smallest such subalgebras, namely, the $C^*$-subalgebra generated by $1,a$ and $a^*$. Then the permanence says if $\lambda-a$ is invertible, then $\lambda-a$ is in the closed linear span of products of $1,a$ and $a^*$ (although order of multiplications matters here and it is not actually a polynomial).

I am wondering whether there is some canonical way to construct these 'polynomials'. That is, given $a\in\mathcal{A}$ invertible, how can one find explicitly the linear span of products of $1,a$ and $a^*$ that converges to $a^{-1}$?

Thanks!
My feeling (not formally justified) is that that there is no canonical choice. Take a look at the simplest example: let 
$$
a=\begin{bmatrix}2&amp;0\\0&amp;3\end{bmatrix},
$$
$\lambda=1$. 
So $a$ is selfadjoint, and of course
$$
(a-\lambda)^{-1}=\begin{bmatrix}1&amp;0\\0&amp;1/2\end{bmatrix}=p(a-\lambda)
$$
for an appropriate polynomial. 
Now, what is the canonical polynomial that takes $1,2$ to $1,1/2$? Let us assume we want the minimum degree possible (i.e. $2$); this is already arbitrary. You could ask that the polynomial be monic, and in this case
$$
p(t)=t^2-\frac72\,t+\frac72;
$$
or you could want $p(0)=0$, in which case
$$
p(t)=-\frac34\,t^2+\frac74\,t.
$$
I don't really see a reason that makes one more canonical than the other.
To answer the question in your title:

If $\lambda -a$ is invertible you can write its inverse as a power series in $a$:

$$\frac{1}{\lambda - a} = \frac{1}{\lambda(1 - \frac{a}{\lambda})} = \sum_{n = 0}^\infty \frac{a^n}{\lambda^{n+1}}.$$