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Chapter 8: Q. 3 (page 669)
What is a power series in x?
A power series in x is.
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What is Lagrange’s form for the remainder? Why is Lagrange’s form usually more useful for analyzing the remainder than the definition of the remainder or the integral provided by Taylor theorem?
In Exercises 49–56 find the Taylor series for the specified function and the given value of . Note: These are the same functions and values as in Exercises 41–48.
Let be a power series in with a finite radius of convergence . Prove that if the series converges absolutely at either , then the series converges absolutely at the other value as well.
If m is a positive integer, how can we find the Maclaurin series for the function if we already know the Maclaurin series for the function f(x)? How do you find the interval of convergence for the new series?
In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder,
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