Prove that under the conditions of the preceding problem, the requirement that \(\sum_{i=1}^{\infty} p_{i} q_{i}=+\infty\) is sufficient not only for the integral theorem but for the local theorem as well.

Preceding Problem:

The probability of occurrence of an event \(A\) in the \(i\) th trial is equal to \(p_{i} ; \mu\) is the number of occurrences of \(A\) in \(n\) independent trials. Prove that

\[
\mathbf{P}\left\{\frac{1-\sum_{k=1}^{n} p_{k}}{\sqrt{\sum_{i=1}^{n} p_{i} q_{i}}}\]

if and only if \(\sum_{i=1}^{\infty} p_{i} q_{i}=\infty\).