Representation of functions in the unit disk by series of rational fractions

It is shown that if $f(z)=\sum_{n=0}^\infty a_nz^n$, $a_n=O(1/n^p)$, $p>1$, then $f(z)$ can be expanded in a series
$$ f(z)=\sum_{k=1}^\infty\frac{A_k}{1-\lambda_kz},\qquad|\lambda_k|<1, $$
that converges uniformly inside the unit disk $|z|<1$. For $p>2$ the expansion is valid in the closed disk $|z|\leqslant1$, and $\sum_{k=1}^\infty|A_k|<\infty$.
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