I have a time series $\{X_t\} = Z_t + \lambda(Z_{t-1} + Z_{t-2} + ... )$, where $Z_t$ is a White Noise process.
Let $\{Y_t\} = X_t - X_{t-1}$
How does $$\{Y_t\} = X_t - X_{t-1} = (Z_t + \lambda\sum_{j=1}^{\infty}Z_{t-j}) - (Z_{t-1} + \lambda\sum_{j=1}^{\infty}Z_{t-1-j}) = Z_t + (\lambda - 1)Z_{t-1}$$
I don't see how those summations can be simplified into $Z_t + (\lambda - 1)Z_{t-1}$