9.3 Power Series: Taylor & Maclaurin Series

Transcription

1 9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a is a power series cetered at a c a a c a c a c 0 1 c, where c is a costat. The equal sig above meas that the left side equals the right side for all values i the domai. This meas the above is a IDENTITY. But for WHAT values of does the idetity hold? We have to fid them. For all such values, we say the series CONVERGES. For the values of for which the idetity is NOT true, we say the series diverges. For a power series cetered at c, eactly oe of the followig is true: 1) The series coverges oly at c. (ALL power series coverge at their ceter!!) ) The series coverges for all. 3) There eists a R 0 such that the series coverges for c R ad diverges for c R. R is called the radius of covergece of the power series. I part 1) the radius is 0. I part ), the radius is I part 3) The correspodig domai, [ c R, c R], is called the iterval of covergece or the domai of the power series. Note: to determie if the edpoits are icluded or ot, we must test each edpoit idepedetly. Note: We typically use the RATIO TEST to determie the radius of covergece. Page 1 of 5

2 Eample 1: 3 4 Based o the fact that a 4 th degree Maclauri polyomial for f e is M4 1,! 3! 4! fid the th term, the fid the radius ad iterval of covergece for the represetative power series. Ofte, we will be dealig with power series represetig ukow fuctios. While we may ot recogize the fuctio the series actually represets, we ca still determie the values of for which it does represet the ukow fuctio. Eample : Fid the radius of covergece ad the iterval of covergece. Be sure to test the edpoits idepedetly. (a) (b) 03 (c) 1 1 (d) 1! 0 0! 3 Page of 5

3 We will ow look at a special family of power series for which you re practically already acquaited: Taylor ad Maclauri Series. Taylor Series cetered at c: f ( c) f ( c) f ( c) f ( ) f ( c) f ( c)( c) ( c) ( c) ( c)!!! Oce agai, if c 0, the series is called a Maclauri series. 0 Notice we ow use a equal sig istead of a approimatio sig. Do you kow why??? Eample 3: Fid a Taylor series for term. 5 f ( ) e cetered at c. Give the first four ozero terms ad the geeral There are four special Maclauri series you must kow. These are the series for e, si, cos, ad 1. These series, uder the operatios of calculus, behave like the fuctios they represet o their 1 iterval of covergece. For series with a fiite iterval of covergece, such as the series for takig the derivative or itegral will ot chage the radius of covergece but may chage the edpoits of the iterval of covergece. 1 1, Oce we have these series memorized these series, we ca coveietly maipulate them to suit other similar o-polyomial fuctios as we did i the previous sectio with Taylor polyomials. You ca maipulate these three special series (or ay series we are give) to fid other series by usig the followig techiques. Note: the radius of covergece may chage, though) 1) Substitute ito a series for ) Multiply or divide the series by a costat ad/or a variable 3) Add or subtract two series 4) Differetiate or itegrate a series (may chage the iterval, but ot the radius of covergece) 5) Recogize the series as the sum of a geometric power series (et sectio) Page 3 of 5

4 Eample 4: If 3! 3!! 1, Fid f ad f 0 f. Do you recogize this familiar fuctio? Eample 5: If f 1 3! 5! 1! you recogize this familiar fuctio?, ad F f d ad F 0 1, fid F. Do Eample 6: Fid a Maclauri series for f ( ) si. Fid the first four ozero terms ad the geeral term. Page 4 of 5

5 Eample 7: Fid a Maclauri series for g( ) cos. Fid the first four ozero terms ad the geeral term. Eample 8: e Fid a Maclauri series for h ( ) e. Fid the first four ozero terms ad the geeral term. Eample 9: Usig kow Maclauri series, prove Euler s Idetity: i e 1 0 Page 5 of 5