Series for $1\mathrm{/}\pi$ arising from Cauchy product

In this note, we evaluate a series for $1\mathrm{/}\pi$ conjectured by Sun. Our proof uses the Cauchy product and hypergeometric transformations. From this result, we derive two additional analogous series for $1\mathrm{/}\pi$ involving polynomials of degree $3$. Further identities can be proved using our method; these are presented in a table at the end of the note.