I don't know too much about formal power series, but by using some common sense
I can say that
$$(1-x)^{-1}=1+x+x^2+x^3+x^4\ldots$$ will only hold iff $\left| x\right| <1$.
Does this series hold iff $\left| x\right| >1$?,\
But controversy is seen in "chapter 9 Renormalization" in Ryder's book on QFT at page 341, its as follows,
$$\begin{align}
D_{\mu\nu}'=D_{\mu\nu}-D_{\mu\alpha}\big(k^\alpha k^\beta-g^{\alpha\beta}k^2\big)\Pi(k^2)D_{\beta\nu}
\end{align}$$
and hence putting $D_{\mu\nu}=-g_{\mu\nu}/k^2,$
$$\begin{align}
D'_{\mu\nu}(k)&=\frac{1}{k^2[1+\Pi(k^2)]}\Bigg(-g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\Pi(k^2)\Bigg)\\&=\frac{-g_{\mu\nu}}{k^2[1+\Pi(k^2)]} + \text{gauge terms}.
\end{align}\tag{9.122}$$...
I tried a lot to attain his final formulae, Finally, I find that,
if $\Pi(k^2)$ is possible to expand as a power series, then I get the desired result,
So I expand
$$(1+\Pi(k^2))^{-1}=1+\Pi(k^2)+(\Pi(k^2))^2+\ldots$$ then I get the result given by Ryder by simple algebraic expansion as follows,
putting $D_{\mu\nu}=-\frac{g_{\mu\nu}}{k^2}$(Feynman's propagator) gives
$$\begin{align}
D'_{\mu\nu}(k)=\frac{-g_{\mu\nu}}{k^2}-\Bigg(\frac{k_\mu k^\nu}{k^2}-g_{\mu\nu}\Bigg)\frac{\Pi(k^2)}{k^2}
=\frac{1}{k^2}\big(1-\Pi(k^2)\big)\Bigg[-g_{\mu\nu}-\frac{k_\mu k^\nu\Pi(k^2)}{1-\Pi(k^2)}\Bigg].\end{align}$$
Iff $\Pi(k^2)<<1$
$$\begin{align}
D'_{\mu\nu}(k)&=\frac{1}{k^2[1+\Pi(k^2)]}\Bigg(-g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\Pi(k^2)\Bigg)\\&=\frac{-g_{\mu\nu}}{k^2[1+\Pi(k^2)]} + \text{gauge terms}.
\end{align}\tag{9.124}$$
But in fact, $\Pi(k^2)>>1$ because $$\Pi(k^2)=\frac{e^2}{6\pi^2\epsilon}+\frac{e^2k^2}{60\pi^2m^2}+\ldots$$, and this $\epsilon$ is very small in fact, its magnitude is almost equal to zero. So we can't use power series in fact, Then how would we say Ryder's derivation is correct??
No; since each term on the RHS is increasing by a factor of $x$, if $\left| x\right|>1$, then $$\lim_{n\to \infty}a_n=\pm \infty,$$ so the series diverges.
Another way to think about it is that since $\left| a_n \right|$ is monotonically increasing, each term "cancels out" the effect of the last one and makes the magnitude grow in either the positive or negative direction. Since the series is infinite, this growth in magnitude never stops, so the series diverges.