Summary: Let \(\{X,X_n,n\geq 1\}\) be a sequence of i.i.d. random variables with \(\operatorname{E}X=0\) and \(0<\operatorname{E}X^{2}=\sigma ^{2}\infty \), and set \(S_{n}=\sum _{i=1}^{n}X_{i}\), \(n\geq 1\). For every \(d>0\) and \(a_{n}=o((\log\log n)^{ - d})\), we show the precise rates in the generalized law of the iterated logarithm for a kind of weighted infinite series of \(\operatorname{P}[|S_{n}|\geq (\epsilon +a_{n})\sigma \sqrt{n}(\log\log n)^d]\).