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1 Taylor Series ad Applicatios - (8.7)(8.8). Taylor ad Maclauri Series: For a give fuctio f!x, how ca we fid its power series represetatio? If f has a power series represetatio at a umber a, that is, if the! b! f!!c This power series is called the Taylor series of f!x at c. Maclauri series. b x # c for x # c R or f!x!! f!!c!x # c. Whe c!, the power series is also called. Taylor s Theorem: Suppose that f has! # derivatives o c # R, c # R for some R $. The for x i c # R, c # R, f!x! P x # R x where P x! f!c # f %!c!!x # c # f %%!c!!x # c #...# f!!c R x! f!#!z! #!!x # c# for some z betwee x ad c.!x # c Example Fid the Taylor series represetatio of f!x at x! ad its iterval of covergece. a.f!x! e x b.f!x! six c.f!x! lx # a. f!x! e x. Set the table:... f!!x e x e x e x e x... e x f!!... b!!...! T!x! # x # x! # x! Fid the iterval of covergece I: use the Ratio Test: R! lim b #! lim & b &! #! So, I!!#,#. Therefore, b. f!x! six. Set the table: e x! # x # x! # x! #...# x #...! lim & #! # R! x #...# #...!! x, for # x. a. e #x b. e x # # x x

2 5... f!!x six cos!x # six # cos!x six cos!x... f!! #... b #!!... T!x! x #! x # x5 #...!!!#! x# #! Iterval of covergece I : R! lim b # & b use the Ratio Test:! lim &!# #! x#! #! #!!# x #! lim &! #! # x!, for all x.# I! #,. Therefore, six! x #! x # x5 #...!!!# x#, for # x.! #! a. si!x b. si x x c. f!x! lx #. Set the table:... f!!x lx #! x#!x # # #!x # #!#!#!x # #!#!!x #... f!! #!#!#!#!... b #!!!#!#... b!, b! #, b!!#,..., b!!# # T!x! x # x # x #...#!# # x #...!!!# # x Fid the iterval of covergece: use the Ratio Test: R! lim b #! lim & b &!# #!## The Taylor series coverges for x. Check two edpoits. Whe x!,!!# # x!! Whe x! #,!!# # x!!!# # So, the iterval of covergece: $#,.! lim &!# #!#! #! #!. harmoic series diverges. alteratig harmoic series coverges. Example Fid thetaylor series represetatio of f!x! Set the table: # x at a! ad the radius of covergece.

3 f!!x f!! c!x # /!!x # #/ #!x # #/ # # #!x # #5/ # #!#!!#!! 5! 5 # # # 5!x # #7/!#!!!5!#!!!5 7! 7 c!!# #!!...! # #.!!##!!...! # # T!x! # x #!#! x #!#!!! 5 x #!#!!!5 x #...! 7 #!# #!!...! # x #... # Fid the iterval of covergece: use the ratio test: lim & c # c!!...! #! #! lim & #! #!! #! lim &! #! # R!. So, the Taylor series coverges for x #. R!. #!!...! # Kow power series represetatios: O page 9: # x!! x! # x # x #... for x e x!! six!! cos!x!! x! # x # x!!# x#!# x # x!# x# l # x!!!# ta #!x!! #... for - $x! #!! x # x! # x5 #... for - $x!! # x! # x #... for - $x!! x # x # x #! x # x # x5 5 #... for # x $ #... for # $ x $ Example Use the kow power series represetatios for e x, six, cos!x, ta #!x ad lx to fid the Taylor Series for e #x/, six, cos!x, l # x, ta # x at a! ad iterval of covergece.

4 e #x /!! six!! # x!#!x #!x cos!x!!!#! l # x!! ta # x!!!# x, for # x! #!!!!# # x #! #!!# #!#x, for # x!!!#! x!!!!#! x, for # x!!!!# # x! # x # x # 8x #..., # x! #! x # x!!!# #, for # $ x $ # # $ x $. Applicatios: Example Express si. as the limit of a sereis. Approximate it by a Taylor polyomial of degree 5 ad estimate the approximatio error. Estimate the umber of terms eeded i a Taylor polyomial to guaratee a accuracy of # Recall:! 5. six!!!#! x# #!!! si.!!!#!. #! #!, for # x! #!!# x# %. #!.! #!.5 R. $.7!. 987 ' 7! #. 5! 8. ' #8! ' # Example Express l.9 as the limit of a sereis. Approximate it by a Taylor polyomial of degree ad estimate the approximatio error. Estimate the umber of terms eeded i a Taylor polyomial to guaratee a accuracy of #. Recall: Let x! #.. l.9!! # x l # x!!!#!# #!#.! #.558!!!#! x # x # x #... for# x $ #.! #. #. #. #.

5 R!x! f!5!z x 5! x 5!! # z 5 # z 5 x5, z is betwee ad. f!x! l # x, f %!x! # x, f %%!x! #! # x, f %%%!x!! # x, f!!x! #! # x, f!5!x!! # x 5 Fid such that R!#. $ #. Let! 9. R!#. $ 5!.5!. ' # R x! f!#!z! #! x!#!! # z #! #! x# $ #!.# $ #! 9, R 9!x $.! #,! 8, R 8!x $.9 9!. ' # Example Evaluate the idefiite itegral & six x dx. & six x dx! & x!!# x#! #! dx! &!!!!#! & #! x dx!!!#! #! # x# # C!# x! #! dx.5 Example Use series to approximate the itegral & cos!x dx to withi three decimal places. cos!x! #!x #!x #...!!!!!#! x &.5 cos!x dx! &.5!!# x! dx!!!# &.5!!!#!! x# #.5!! x! dx!#.5#!! # Fid m such that R m $.5. R m $ b m#!.5 m#!m #!!m # Whe m! 5,.5!!!!. 9 ' #. Whe m!,.5 9!8!!9! 5. 8 ' #9. Whe m!,.5 7!!!7!. 55 ' #. Whe m!,.5 5!. ' #!!!5 Let m! 5. Example Approximate e # ad! correct to decimal places. 5

6 e #! # #! #! #...!! ta #!! # # 5 #!!!# # Fid m such that R m $.5. R m $ b m# $ m # $.5 # m # '.5! m '! #! Let m! Example Fid the sum of the series. a.! x!x!!! e x a.! x b.!!#!! c.! x #!# b.!!!!#!!!!! cos! c.! x #!x! x!! x e x # # x Example Fid the Taylor series of f!x! # x at a! ad the radius of covergece. Sketch the graphs of f!x ad T m!x for m!,,,. Approximate. to decimal places. Approximate # x dx to 5 decimal places. &. Set the table: f!!x f!! c!x # /!x # #/ #!x # #/ #!#!!! # #!x # #5/ # #!#!!! # # # 5!x # #7/!#!!!5!#!!!5! : : : : # #... # #!x # #7/!# #!!...! #!# #!!...! # T!x! # x #! x #!! x...#!# #!!...! #! x #...! # x #!!# #!!...! # x

7 Check the radius of covergece: lim & c # c! lim &!!...! # #! #!! lim & #! #! # R!!!...! #.! f!.! T!.! #!. #!!#!!...! #!.!!...!m # This is a alteratig series. R m $ b m#! m#!. m#.5!m #! Whe m!,!! #! #!!.#!. 5 ' #5 Whe m!,! #! #!!.#!. 5 ' # $.5. Let m!.. % #!. #!!.!. 875.! usig the Scietific Notebook.. & # x. dx! & # x #!!#!!...! #!x dx! x # 8 x #!!# #!!...! #!!. # 8!. #!!# #!!...! # Fid m such that R m $ b m#.5.!!...!m # R m $ b m#! m#!m #!!! Whe m!, #! #!! Whe m!, #! #! Let m!. &. &. m #!.m#. # x#! #!.!#!. 5 ' #! #!.!#! ' #9 Whe m!, 8!.!. 5 ' #5 $.5 # x dx %!. # 8!.!.5 # x dx!.98 usig the Scietific Notebook. #!.# 7