By AM-HM inequality, $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} \geq \frac{M^2}{f(1) + f(M+1) + 2(f(2) + ... + f(M))} = \frac{M^2}{2(f(1) + ... + f(M+1)) - f(1) - f(M+1)} = \frac{M^2}{(M+1)(M+2) - f(1) - f(M+1)} \geq \frac{M^2}{(M+1)(M+2) - 1 - 2} = \frac{M^2}{M^2 + 3M - 1} > \frac{M^2}{M^2 + 3M} = \frac{M}{M + 3}$