The sum to $n$ terms of the series $1 + 2\left (1 + \dfrac{1}{n}\right) + 3{\left (1 + \dfrac{1}{n}\right)^2} + ...$ is given by

The series is an arithmetico-geometric series: 1 + 2r + 3r^2 + ... + nr^(n-1) where r = 1 + 1/n. The sum S = 1 + 2r + 3r^2 + ... + nr^(n-1). Then rS = r + 2r^2 + ... + (n-1)r^(n-1) + nr^n. Subtracting gives (1-r)S = 1 + r + r^2 + ... + r^(n-1) - nr^n = (r^n - 1)/(r-1) - nr^n. With r-1 = 1/n, we get (-1/n)S = ( (1+1/n)^n - 1 ) / (1/n) - n(1+1/n)^n. This simplifies to S = n^2.