Sign-oscillations for power series with random coefficients

We can construct inductively a sequence $t_n \to 1-$ and an increasing sequence $K_n$ of positive integers such that with probability $> 1 - 1/n^2$, $f(t_n)$ has the same sign as $V_n = \sum_{k=K_{n-1}+1}^{K_n} a_k t_n^k$. Since $\sum 1/n^2 < \infty$, almost surely $f(t_n)$ has the same sign as $V_n$ for all but finitely many $n$. Almost surely the independent random variables $V_n$ change sign infinitely often, so a.s. $f(t_n)$ changes sign infinitely often.