# Wold Decomposition of an AR(2) process: Expanding a product of geometric series

In the last step of this answer, the author writes that to obtain Wold Representation of an AR(2) process, you need to expand the geometric sequences in the fractions on the right hand side:

$$X_{t}=\frac{1}{(1-\lambda_{1}L)}\frac{1}{(1-\lambda_{2}L)}\varepsilon_{t}$$

After expanding the sequences I am stuck with the following:

$$X_{t}=(1+\lambda_{1}L+\lambda_{1}^{2}L^{2}+...+\lambda_{1}^{n}L^{n})(1+\lambda_{2}L+\lambda_{2}^{2}L^{2}+...+\lambda_{2}^{n}L^{n})\varepsilon_{t}$$

How can I get rid of the lag operators and get to the MA representation from here?

\begin{align} \lambda(L) &= (1+\lambda_{1}L)^{-1} (1+\lambda_{2}L)^{-1} \\[12pt] &= (1+\lambda_{1}L+\lambda_{1}^{2}L^{2}+...+\lambda_{1}^{n}L^{n})(1+\lambda_{2}L+\lambda_{2}^{2}L^{2}+...+\lambda_{2}^{n}L^{n}) \\[12pt] &= 1 + (\lambda_1+\lambda_2) L + (\lambda_1^2 + \lambda_1 \lambda_2 + \lambda_2^2) L^2 + (\lambda_1^3 + \lambda_1^2 \lambda_2 + \lambda_1 \lambda_2^2 + \lambda_2^3) L^3 + \cdots \\[6pt] &= \sum_{i=0}^\infty \Bigg( \sum_{j=0}^i \lambda_1^{i-j} \lambda_2^j \Bigg) L^i. \\[6pt] \end{align}
\begin{align} X_t = \lambda(L) \varepsilon_t &= \sum_{i=0}^\infty \Bigg( \sum_{j=0}^i \lambda_1^{i-j} \lambda_2^j \Bigg) L^i \varepsilon_t \\[6pt] &= \sum_{i=0}^\infty \Bigg( \sum_{j=0}^i \lambda_1^{i-j} \lambda_2^j \Bigg) \varepsilon_{t-i}. \\[6pt] \end{align}