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Video 13 - Ratio and Root Tests, Power Series, Taylor Series.
- Absolute Convergence and Alternating Series.
Absolute convergence is reviewed as well as forms of the comparison test and limit comparison for series with negative as well as positive terms in the determination of absolute convergence. Conditional convergence is reviewed. Alternating series are defined and the nth term test for convergence of an alternating series is reviewed. Examples illustrating the concepts are worked as well as examples using the nth term to estimate the error in a partial sum and examples of finding the proper partial sum for estimating an alternating series sum to within predetermined error tolerance.
- Ratio Test and Root Test.
The ratio test and the root test for determining convergence or divergence of infinite series are reviewed and discussed. Examples are worked illustrating both tests as well as how to choose between the two n tests from the form of the nth term.
- Power Series.
Power series, radius of convergence, and interval of convergence are defined and discussed as well as the theorems on termwise differentiation and integration of power series. The ratio and root test forms for determining radius of convergence are reviewed and examples are worked illustrating their use.
- Taylor Series.
The Taylor series of a function is defined and the Lagrange form of the Taylor remainder is reviewed and used to show certain functions equal their Taylor series. Examples are also worked illustrating techniques of algebra combined with termwise differentiation and integration to obtain Taylor series of certain functions from the formula for the sum of the geometric series.
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