Applications of a class of transformations of complex sequences

Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi }$ due to Ramanujan. We then apply Mishev’s transform to provide proofs for a number of related hypergeometric identities, including a new and simplified proof for a family of series for $\frac{1}{\pi }$ previously obtained by Levrie via Fourier–Legendre theory. We generalize this result using Mishev’s transform, so as to extend a result due to Guillera on a Ramanujan-like series involving cubed binomial coefficients and harmonic numbers.