10. (a) Find a function Whose Taylor series is 200 in 11:1 H (b) State its interval of convergence (c) Use your result above to evaluate the series 220:1 %,ln a ) function whose taylor series is is f ( x ) : e " - 1 n = 1 explanation Taylor series of ex - 2 ( 2 7 ) n =0 ". to obtain serves starting from ms1, we can simply subtract mso team which ind : . Taylor servies is S ( x ) is 8 /x ) = e - 1 n = 1 b ) The interval of convergence of the taylor series of f (x ) is the set of all x for which shes Converges . By ratio test , we have . ( n + 1 ) n converges for all x which means interval of convergence is ( - , D )( ) The series M (- 1) is an alternating n = on Series . By alternating servies test , the Servies converges if absolute value of its terms forms a decreasing sequence that Converges to zero . We have : not 1 - 1 ) (it ) ) ! Therefor the sequence of absolute values is decreasing . Moreover, we have : -1 ) which converges to zero as mood . Therefore the series . By the alternating n ! serves exxon bound , we have : C - - L / S ( - 1 ) where is n ( N+ 1 ) ' Limit of series . Since L= e -1 m wehave L - 1 - (e - 1 ) / 5 e - 1