12.5.17 Olympic algebra

condition: Let \( f(x)=1+a_{1} x+a_{2} x^{2}+\cdots \) ​​and let all coefficients in the expansion of the relation \( f^{\prime}(x) / f(x) \) in powers \( x \) do not exceed 2 in modulus. Prove that \( \left|a_{n}\right| \leq n+1 \).