Taylor Series 1

# Taylor Series 1 - Analysis WebNotes Chapter 07 Class 45...

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Analysis WebNotes: Chapter 07, Class 45 1 of 4 11/9/2007 11:19 AM Class Contents Taylor series General power series Higher order derivatives Differentiating power series Power series are unique Taylor Series General Power Series All of the power series that we have studied so far have been of the form and we showed that they converge inside disks centered on zero, and diverge outside those disks. However, we can easily do a change of variable, to study power series of the form where z 0 is fixed. This series converges on a set of the form which is a disk of radius R centered on z 0 . In this section we shall be interested in real-valued functions of a real variable which come from power series. The power series will be of the form where x and x 0 are real numbers. The region in the real line where this series converges is the interval (x 0-R , x 0+R ) , which is called the interval of convergence (it is just the intersection of the disk of radius R centered on x 0 with the real line).

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