How long does Google indexes the website after algorithm update?
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I've noticed that the page ranks for some queries were improved unexpectedly, without any actions from my side. Is it possible that this improvement is connected with with Google algorithm update Sep, 4-5?
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That's possible but really very difficult to answer with certainty. One of those things you'll need to accept that you'll really never know

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It could be, but if you don't have any crawl data available from your log files you likely can't say with certainty that the two are related.