微积分教学课件chapter11.10.ppt

Example: Question: Under what circumstances is a function equal to the sum of its Taylor series ? In general, f(x) is the sum of its Taylor series iff It is often helpful to make use of the following fact: Example: Example: Example: Example: Find the Maclaurin series for sinx and prove that it represents sinx for all x. Example: Example: Example: For future reference: Example: (b) Example: Multiplication and Division of Power Series (b) * In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: which functions have power series representations? How can we find such representations? 8.7 Taylor and Maclaurin Series Maclaurin Series. Solution: We start by supposing that f is any function that can be represented by a power series ............. (1) ............. (2) ............. (3) ............. (4) we get then In the case of the Taylor series, the partial sums are: Let If we can show that Then: we often use: Can you prove the result? Solution: Proof: Choose any positive number d such that Then: That is: In particular: Solution: Solution: The Maclaurin series is: So: In summary: Solution: That is: Solution: Solution: …………… We can show that the remainder tend to 0 as n becomes arbitrarily large. Then: Solution: (a) According to the alternating theorem, we know that the error is smaller than Solution: *