Math 120 Answers for Homework 23

Transcription

1 Math 0 Aswers for Homewor. (a) The Taylor series for cos(x) aroud a 0 is cos(x) x! + x4 4! x6 6! + x8 8! x0 0! + ( ) ()! x ( ) π ( ) ad so the series ()! ()! (π) is just the series for cos(x) evaluated at x π. The value of the ifiite sum is therefore cos(π). (b) The Taylor series for e x aroud a 0 is ad so the series e x + x + x! + x! + x4 4! + x5 5! + x6 6! + x x. The value of the ifiite sum is therefore e e. (c) The Taylor series for l( + x) aroud a 0 is ad so the series ( ) is the series for e x evaluated at l( + x) x x + x x4 4 + x5 5 x6 6 + ( ) + 5 ( ) + ( ) + x ( ) is the series for l( + x) 5 evaluated at x 5. The value of the ifiite sum is therefore l( + 5 ). (d) The series + ( ) + is the power series + x+ evaluated at x, but exactly what fuctio that power series describes is ot immediately clear. If we set f(x) + x+ the differetiatig we fid that

2 f (x) + ( + ) x x x. where the last equality follows from recogizig the series as a geometric series with commo ratio x. This tells us that f(x) is a atiderivative for x. Sice x dx ( + x + ) dx (l( + x) l( x)) + C, x we see that f(x) (l( + x) l( x)) + C l ( +x x) + C. Pluggig i x 0 ito the origial series, we have f(0) + (0)+ 0. Pluggig i x 0 ito the descriptio of f above we have f(0) l ( +0 0) +C l() + C C ad therefore that C 0, i.e., we see that f(x) l ( +x x). ( ) The value of the ifiite sum is therefore f( ) l + l() l( ).. (a) By the biomial series theorem the series for ( + x) aroud a 0 is ( + x) 0 x + x + ( )( 4)! x + ( )( 4)( 5) x! + ( )( 4)( 5)( 6) 4! x +! x 60! x ! x4 50 x 5 + 5! This series coverges wheever x <. x 4 + ( )( 4)( 5)( 6)( 7) x 5 + 5!

3 (b) For all x <, + x ( ) x x+x x +x 4 x 5 + x 6, by the formula for the sum of a geometric series. (c) Differetiatig the equality from (b) twice, we fid that ( + x) ( ) ( )x x + x 0x + 0x 4 or ( + x) ( ) ( )x ( + )( + ) ( ) 0 x x + 6x 0x + 5x 4 which is the same as the series from (a). To be certai that these series really are the same (ad do t just happe to match for the first few terms) ote that ( )( 4)( 5) ( )! ( ) ( + )( + ).. ( ) 4 5 ( + ) ( + ) 4 ( ). The series for e x ad si(x) are oes we already ow, so: + si(x) + e x x + x + x! + x! + x4 4! + x5 5! + x6 6! + ( ) ( + )! x+ + x x! + x5 5! x7 7! + x9 9! x! + while the series for x ( x) is somethig we ca wor out from the biomial series theorem:

4 ( x) 0 ( x) + ( x) + ( )( ) ( x) +! + x + x + 5 x + Comparig coefficiets of the Taylor expasios up to the x term we see that ear x 0 (i.e. for small x), the fuctios satisfy the iequalities + si(x) e x x. The graphs of the three fuctios cofirm what we ca see from the Taylor series. At x 0 all three fuctios have the same value ad the same taget lie (this is what we ca read off from the c 0 ad c coefficets of the Taylor series). y x y e x y + si(x) However, ear x 0 the fuctio + si(x) curves dowwards from the taget lie (sice the ext ozero coefficiet, that of x, is egative i its Taylor expasio) while both x ad e x curve upwards, sice the coefficiets for x i both expasios are positive. Furthermore, the graph of x curves upwards faster tha that of e x, sice the coefficiet of x i the expasio of x is larger tha the coefficiet of x i the expasio of e x. 4. By the biomial series theorem, for x < we have Therefore, ( x ) 0 0 ( x ) ( ( ) arcsi(x) ( ) )x 0 dx ( ) x. 0 0 ( ) ( ) x + + C. + (( ) )x dx 4

5 Sice arcsi(0) 0 we see that the costat term is C 0 ad so ( ) arcsi(x) ( ) x This formula ca be writte i a slightly more coveiet way. Whe ( )( )( 5) ( )! so the formula for arcsi(x) above is 5 ( ) ( )! arcsi(x) ( ) x + ( + )! ( )! ( + ) ( )!! x+ ()! ( + ) (!) x+ 0 ()! ( + ) (!) x+. 5. Suppose that f(x) is a ifiitely differetiable fuctio defied o all of R ad that f () (0) for all 0. (a) The coefficiets of the Taylor series c x for f(x) aroud a 0 are give by the formula c f() (0), so c f() (0) ( )! ad the Taylor series is ( )! x. (b) The -th coefficiet of the power series is c covergece is R lim c c + lim ( )! ( )! ad therefore the radius of + lim Therefore the series has a ifiite radius of covergece. 5 +.

6 (c) Sice the Taylor series has a ifiite radius of covergece if we plug i ay value of x we get a ifiite sum which coverges to a umber. The assertio that f(x) ( )! x is the assertio that for every value of x the value of that ifiite sum is the umber f(x). We certaily used this id of equality i this homewor assigmet: without it we would t have bee able to compute the values of the ifiite sums i the first questio. { } (d) If M + max f (+) (x) x [, ], the the corollary to Taylor s theorem tells us that for ay x 0 [, ], ad ay 0, where P (x) ( )! x 0. 0 f(x 0 ) P (x 0 ) M + x 0 + ( + )!. ( )! x 0 is the -th Taylor polyomial for the ifiite series O the iterval [, ] the largest possible value of x 0 is, so the corollary also gives us the iequality + f(x 0 ) P (x 0 ) M + ( + )!. We ow that M + 40( + ), ad usig that i the iequality above gives us: f(x 0 ) P (x 0 ) 40( + ) + ( + )! 40 + ( )!. Sayig that the value of the ifiite series is the the umber f(x 0 ) is the same thig as sayig that the differece above goes to zero as goes to ifiity. + Sice lim 0, the squeeze theorem ad the iequality above give us ( )! lim f(x 0 ) P (x 0 ) 0, ad so f(x 0 ) ( )! x 0, i.e., o the iterval [, ] we have the equality of fuctios f(x) ( )! x. 6