EMA and MACD weights

With an infinite history, let $E_t=\alpha\sum_{k\ge0}(1-\alpha)^kp_{t-k}$ be the EMA of log prices. (a) Prove $p_t-E_t=\sum_{i\ge0}(1-\alpha)^{i+1}r_{t-i}$. (b) Hence write $\mathrm{MACD}_t=E^{(f)}_t-E^{(s)}_t$ (fast $\alpha_f$ > slow $\alpha_s$) as $\sum_i v_ir_{t-i}$, show $v_i>0$, and find $\sum_iv_i$. (c) For the conventional 12/26 EMAs ($\alpha=2/(n+1)$), find the lag at which $v_i$ peaks.