Wikipedia:Reference desk/Archives/Mathematics/2012 January 9

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January 9

series acceleration

hello. What is the best form of series acceleration to apply to the (real-valued) Taylor series for e^x (which already converges pretty fast, but hey)? I had a glance at your series acceleration article but it doesn't seem to say which is best used when. This is preferably something that it is easy to write a computer program for. Thanks. 24.92.85.35 (talk) 22:56, 9 January 2012 (UTC)

I don't think there are many specific rules since it depends on factors independent of the specific series. For example if you're only doing a few values to a relatively low accuracy and the series converges reasonably quickly then it may be simpler just to go with the original series. Otherwise it's generally a tradeoff between accuracy, speed of convergence and how complex of an algorithm you're willing to program. The series for e^x already converges pretty rapidly unless x is large, and there are simple identities you can apply to reduce to the case where x is within given bounds. Instead of trying to accelerate the series it might be better to use a different method such as continued fractions to get more accuracy, but in many cases adding a few more terms to the series is just as effective.--RDBury (talk) 14:44, 10 January 2012 (UTC)