微积分教学课件chapter8.6.ppt

Example: Example: Example: Example: The interval of convergence of a power series is the interval that consists of all values of x for which the series converges. For the inverval of convergence, in case (iii) there are four possibilities: Example: Example: For example: For future reference: Example: Example: Example: Differentiation and Integration of Power Series Notes: Example: Example: Example: Example: Example: (b) Bessel function: Solution: That is: The Radius of convergence is R=1 Solution: ,we get C=0 Set x=1/2,we have Solution: ,we get C=0 The Radius of convergence is R=1 * 8.6 Power Series Solution: Thus the given series converges only when x=0. 0 Solution: ,for what values of is it convergent ? For what values of is the series convergent ? Solution: diverges converges Thus the given power series converges for Solution: The Ratio Test gives no information when so we must consider When the series is converges If the series is diverges Thus the given power series converges for 3 2 4 By convention: Solution: The radius of convergence is diverges converges Solution: The Radius of convergence is R=3 When x=1, the series is diverges diverges Example: Find the radius of convergence of the series Solution: It converges if that is It diverges if that is Thus the radius of convergence is When the series is diverges When the series is diverges And when it converges ,it can be represented by So The sum of the series is a function Test the series for convergence or divergence Solution: Since when so when We can test the series by Integral Test. The function is continuous,positive ,and decreasing on Because Thus , is convergent and the series is convergent. Therefore the series is convergent by Comparison Test. Solution: Solution: So, the interval of convergence is (-2,2). Solution: That is Example: Find a pow