I am absolutely new to trading and would like to start to learn how indicators work. I would like to start with EMA. I found a following formula somewhere on the internet:

EMA_tod = P_tod*K + EMA_yest*(1-K), where K=2/(N+1), N=the number of days in the EMA, P_tod=today's price, EMA_yest= the EMA of yesterday. The problem for me is that I can't see anything exponential in that formula. I think, it's more weighted moving average. Shouldn't exponential moving average look somehow like this?

EMA_today = K*(p1 + (1-K)*p2 + (1-K)^2*p3 + (1-K)^3*p4 + ...), where p1 is today's price, p2 yesterday's price and so on. Here I can easily see that older data are exponentially smaller than the latest ones. Could anybody explain me how it should be? I would like to understand it, as it belongs among basic indicators. Sorry for my poor English.