A property of entire functions with real taylor coefficients

Suppose that $f(z)$ is an entire transcendental function with real Taylor coefficients, $M(r)=max|f(z)|$ on $|z|=r$, and $\{\lambda_n\}$ is the sequence of sign changes of the coefficients. We will show that if $\sum(1/\lambda_n)<\infty$, then $\overline{\lim\limits_{r\to\infty}}\ln\cdot|f(r)|/\ln M(r)=1$.