Question

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " A geometric series with ratio \(r\) converges if \(|r|<1\)

Short answer

'If \(|r|<1\), then a geometric series with ratio \(r\) converges.'

Step-by-step solution

Identify P and Q

To convert the sentence into the given form, we first need to define what \(P\) and \(Q\) are. Considering the original sentence, 'A geometric series with ratio \(r\) converges if \(|r|<1\)', we can see that the condition (\(|r|<1\)) will be our 'If' clause (\(P\)), and the outcome ('A geometric series with ratio \(r\) converges') will be our 'then' clause (\(Q\)).

Formulate the Conditional Statement

Now insert \(P\) and \(Q\) into the conditional sentence structure 'If \(P\), then \(Q\)'. This gives us our final statement: 'If \(|r|<1\), then a geometric series with ratio \(r\) converges.'