“If my Google PageRank moves up to a 3 and my competitor’s PageRank remains a 1, will that push me above them in Google’s search engine rankings?”
Unfortunately, the answer is no. PageRank and your SERPs are not related in that sense. PageRank is an authority number assigned by Google based on an algorithm associating several factors to determine your site’s trustworthiness which, indirectly affecting your rankings in Google for specific keyword phrases. It is not used by Google to determine your rankings for keywords. It is amazing, that in 2009 that some webmasters, business owners and marketers still put emphasis on Google PageRank when determining the goals of your search engine optimization efforts. The goal of your SEO campaign should be to increase relevant visitors to your website over time from the search engines.
That being said, Google does use some of the same factors in its ranking algorithm as it does in its PageRank algorithm. But there are factors used to determine keyword rankings that are not used in PageRank. Keyword placement in your URL for example, is a factor that Google may use for search engine ranking purposes, but it doesn’t affect your PageRank at all. Other factors such as quality content, internal linking, etc do not affect PageRank, but are used to determine your placement on the SERPs.
Bottom line, don’t expect advances in your PageRank to affect your search engine rankings. The two are not related at all…and focus your energy on marketing your website and business online and to become an authority in the eyes of your visitors and the search engines.