  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.
