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For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $\sum_{n=1}^\infty a_n^2<\infty$ and $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned condition on $(a_n)$, this series will indeed converge because eventually (for some natural $N$ and all $n\ge N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.

For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $\sum_{n=1}^\infty a_n^2<\infty$ and $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned conditions on $(a_n)$, this series will indeed converge because eventually (for some natural $N$ and all $n\ge N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.

For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned condition on $(a_n)$, this series will indeed converge because eventually (for some natural $N$ and all $n\ge N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.

For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $\sum_{n=1}^\infty a_n^2<\infty$ and $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned condition on $(a_n)$, this series will indeed converge because eventually (for some natural $N$ and all $n\ge N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.

For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned condition on $(a_n)$, this series will indeed converge because eventually (for all $n\ge N$ for some natural $N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.

For $$S^2 = T + 2 \sum_{n=1}^\infty a_n w_n \mbox{ with } w_n=\sum_{m=n+1}^\infty a_m\tag{1}$$ to be valid, it is enough that $\sum_n a_n$ be any alternating series with $|a_n|$ converging to $0$ (eventually) monotonically.

Indeed, since (1) holds for finite sums, this question boils down to the convergence of the series $\sum_{n=1}^\infty a_n w_n$. Given the mentioned condition on $(a_n)$, this series will indeed converge because eventually (for some natural $N$ and all $n\ge N$) we will have $|w_n|\le|a_{n+1}|\le|a_n|$ and hence $\sum_{n=N}^\infty |a_n w_n|\le\sum_{n=N}^\infty a_n^2<\infty$.