It is indeed true that for a nuclear Frechet space $F$, the complete projective tensor product $F'_\beta \tilde{\otimes}_\pi F$ is isomorphic to $L_\beta(F,F)$. The mistake is your claim that the complete projective tensor product $E\tilde{\otimes}_\pi F$ consists of convergent series $\sum\limits_{n=1}^\infty \lambda_n e_n \otimes f_n$. This is true if $E$ and $F$ are both Frechet spaces or both strong duals of Frechet spaces. In the mixed case as in your situation the complete injective tensor product can be described by such series but it is much smaller than the projective tensor product.

It is indeed true that for a nuclear Frechet space $F$, the complete projective tensor product $F'_\beta \tilde{\otimes}_\pi F$ is isomorphic to $L_\beta(F,F)$. The mistake is your claim that the complete projective tensor product $E\tilde{\otimes}_\pi F$ consists of convergent series $\sum\limits_{n=1}^\infty \lambda_n e_n \otimes f_n$. This is true if $E$ and $F$ are both Frechet spaces or both strong duals of Frechet spaces. In the mixed case as in your situation the complete inductive tensor product can be described by such series but it is much smaller than the projective tensor product.

EDIT: I meant the inductive tensor product (the injective coincides indeed with the projective one because of nuclearity).