Average signal power (signals and systems - discrete signal)
The second formula you have posted gives the average power for an interval from -N to N as N tends to infinity. Using this formula, as you have stated the summation of individual 1's from -N to N, whatever N is,will just be 2N+1, and so 2N+1/2N+1=1 and so the limit is just 1,as N has been eliminated from the expression.