Covergece: th-term Test, Comparig No-egative Series, Ratio Test
Power Series ad Covergece We have writte statemets like: l + x = x x2 + x3 2 3 + x + But we have ot talked i depth about what values of x make the idetity true. Example: Ivestigate whether or ot x = 0.5 makes the setece above true? x = 00? x = 0.5: l +.5 =.5.52 2 +.53 3 +.5 + Appears to Coverge. x = 00: l + 00 = 00 002 2 + 003 3 + 00 + Appears to Diverge. We eed better ways to determie the values of x that make the series coverge.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for x : + x + x 2 + x 3 + + x + Ivestigate the partial sums of the series ad compare the results to y = x y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for x : + x + x 2 + x 3 + + x + Ivestigate the partial sums of the series ad compare the results to y = x y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for x : + x + x 2 + x 3 + + x + Ivestigate the partial sums of the series ad compare the results to y = x y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for x : + x + x 2 + x 3 + + x + Ivestigate the partial sums of the series ad compare the results to y = x y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for x : + x + x 2 + x 3 + + x + Ivestigate the partial sums of the series ad compare the results to y = x y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for x : + x + x 2 + x 3 + + x + Ivestigate the partial sums of the series ad compare the results to y = x y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for : x + x + x 2 + x 3 + + x + O (, ), {y 2, y 3, y 4, y 5, y 6, } coverges to y =. The x polyomial series is a good approximatio of y o (, ). y = x y 2 = y 3 = + x y 4 = + x + x 2 y 5 = + x + x 2 + x 3 y 6 = + x + x 2 + x 3 + x 4 Widow: 2 x 2 ad 0 y 0
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: x x3 5! + + x 2+ 2 +! Ivestigate the partial sums of the series ad compare the results to y = si x o a graph. y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: x x3 5! + + x 2+ 2 +! Ivestigate the partial sums of the series ad compare the results to y = si x o a graph. y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: x x3 5! + + x 2+ 2 +! Ivestigate the partial sums of the series ad compare the results to y = si x o a graph. y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: x x3 5! + + x 2+ 2 +! Ivestigate the partial sums of the series ad compare the results to y = si x o a graph. y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: x x3 5! + + x 2+ 2 +! Ivestigate the partial sums of the series ad compare the results to y = si x o a graph. y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: x x3 5! + + x 2+ 2 +! Ivestigate the partial sums of the series ad compare the results to y = si x o a graph. y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for si x: y = si x y 2 = x y 3 = x x3 3! y 4 = x x3 5! y 5 = x x3 5! x7 7! y 6 = x x3 x x3 5! + + x 2+ 2 +! O (, ), {y 2, y 3, y 4, y 5, y 6, } coverges to y = si x. The polyomial series is a good approximatio of y o,. 5! x7 7! + x9 9! Widow: 7 x 7 ad 5 y 5
Ivestigate the error fuctio: Provig Sie Coverges x2+ Prove the Maclauri series k=0 coverges to si x for all real x. R x = g + c x 0 + +! R x = g + c x 0 + +! = g + c x + +! x+ +! x + +! 2+! Ivestigate the limit of the last statemet lim x + +! = 0 This meas that R x 0 for all x. Therefore the Maclauri series coverges for all x.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 Ivestigate the partial sums of the series ad compare the results to y = e x2 x 0 0 x = 0 y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 7 x 7 ad 5 y 5 o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 Ivestigate the partial sums of the series ad compare the results to y = e x2 x 0 0 x = 0 y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 5 x 5 ad 0 y o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 Ivestigate the partial sums of the series ad compare the results to y = e x2 x 0 0 x = 0 y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 5 x 5 ad 0 y o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 Ivestigate the partial sums of the series ad compare the results to y = e x2 x 0 0 x = 0 y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 5 x 5 ad 0 y o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 Ivestigate the partial sums of the series ad compare the results to y = e x2 x 0 0 x = 0 y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 5 x 5 ad 0 y o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 Ivestigate the partial sums of the series ad compare the results to y = e x2 x 0 0 x = 0 y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 5 x 5 ad 0 y o a graph.
Whe is a Taylor Series a Good Approximatio? Cosider the Maclauri Series for e x2 x 0 0 x = 0 : 0 O 0, {y 2, y 3, y 4, y 5, y 6, } coverges to y = e x2 x 0 0 x = 0. The series is a good approximatio of y o 0. y = e x2 x 0 0 x = 0 y 2 = 0 y 3 = 0 y 4 = 0 y 5 = 0 y 6 = 0 Widow: 5 x 5 ad 0 y
The Covergece Theorem for Power Series There are three possibilities for respect to covergece: =0 c x a with Radius of Covergece. There is a positive umber R such that the series diverges for x a > R but coverges for x a < R. The series may or may ot coverge at either of the edpoits x = a R ad x = a + R. 2. The series coverges for every x (R = ). 3. The series coverges at x = a ad diverges elsewhere (R = 0). The set of all values of x for which the series coverges is the iterval of covergece.
The -th Term Test If lim a 0, the the ifiite series = a diverges. OR If the ifiite series = a coverges, the lim a = 0. If lim a = 0, the ifiite series Whe determiig if a series coverges, always use this test first! The coverse of this statemet is NOT true. = a does ot ecessarily coverge.
Just because the -th term goes to zero does ot mea the series ecessarily coverges. We eed more tests to determie if a series coverges. Examples Use the -th Term Test to ivestigate the covergece of the series.. = 2+ 2 + lim = 2 Sice the limit is ot 0, the series must diverge. 2. = 5 0 lim 5 0 = 0 Sice the limit is 0, the -th Term Test is icoclusive.
The Fiey Procedure for Determiig Covergece th-term Test Is lim a = 0?? Yes or Maybe No The series diverges.
Coverget Geometric Series The geometric series = ar coverges if ad oly if r <. If the series coverges, its sum is a. r Example: Determie if Where a is the first term ad r is the costat ratio. = ( e π ) coverges. This is a geometric series with r = e π. Sice e π <, the series coverges. This explaatio would warrat full credit o the AP Test. Statemets like I kow this is a series with, so it diverges/coverges are acceptable.
The Fiey Procedure for Determiig Covergece th-term Test Is lim a = 0? Yes or Maybe Geometric Series Test Is a = a + ar + ar 2 +?? No No Yes The series diverges. Coverges to if r <. a r Diverges if r >.
a Ivestigate the series diverges. =0! 2 2! Example! 2 =0 to see if it 2! Notice the series is NOT geometric. = + 2 + 6 + 20 + 70 + 252 + Usig partial sums, it appears the series coverges: ;.5;.667;.77;.72 Similar to a Geometric Series, we ca ivestigate the ratio betwee terms. 2 a 0 a2 0.5; 6 a 2 a3 20 0.3; a 2 6 a4 70 0.33; a 20 3 0.29; a 5 a 4 252 70 0.28 Similar to a covergig Geometric Series, the limit of the commo ratio appears to be less tha. Accordig to the ext theorem. This meas the series coverges.
Suppose the limit lim or is ifiite. The: The Ratio Test a + a = L either exists. If L <, the series = a coverges. 2. If L >, the series = a diverges. 3. If L =, the test is icoclusive. Each successive terms are gettig smaller. Each successive terms are gettig larger. This is the cousi test to determiig if a Coverget Geometric Series test. It is idetical except: () If the commo ratio is, the test fails. (2) The test does ot determie the value the series coverges to.
Ivestigate the series Example! 2 =0 to see if it diverges. 2! Similar to a Geometric Series, we ca ivestigate the ratio betwee terms. lim a a lim lim! 2 2 2!! 2! 2 222 2 2! 22!! 2! lim lim 2 2 2 4 62 4 Similar to a covergig Geometric Series, this commo ratio is less tha. Accordig to the Ratio Test, this meas the series coverges.
Notice: Ulike the last example, this series depeds o a value of x. Example 2 lim I #3 o Taylor Series Challeges we worked with the series x 2 2. Fid the radius of covergece. = 3 2 2 a a lim 2 x 2 2 3 2 2 3 2 22 x x 2 lim 3 2 2 3 22 2 2 2 2 x 2 The limit depeds o. Separate the s. x 2 x 2 2 x 2 lim lim 3 2 3 9 2 2 2 If the series coverges, the ratio is less tha. x2 2 x 2 9 2 9 x 2 3 The iterval of covergece is cetered at 2 with a radius covergece of 3.
Example 3 Fid the radius of covergece for the Maclauri series for f x = e x2. We kow: lim Thus: Ivestigate the ratio: a a e x = + x + x2 2! + x3 3! + + x! + = e x2 = + x 2 + x4 2! + x6 3! + + x2! + = lim x 2 2! x! lim The series has a ifiite radius. 22 x!! x 2 =0 =0 x! x 2! lim 2 x 0
Example 4 Fid the radius of covergece for the series Ivestigate the ratio: lim a a Sice the ratio tests uses a absolute value, the powers o - do ot affect the limit. lim 5 x lim 5 5 5 x x lim x 5 x x =. lim 5 x 5 5 5 x x If the series coverges, the ratio is less tha. x 5 x 5 The iterval of covergece is cetered at 0 with a radius covergece of 5.
White Board Challege Use the Ratio Test to determie the radius of + covergece for x =. lim a a lim x x lim 2 lim 2 The Ratio Test is icoclusive because the limit is. We eed to use other tests to determie if the series coverges.
The Direct Compariso Test Suppose 0 a b for all N. If If Both series must have oly positive terms. = b coverges, the = a coverges. = a diverges, the = b diverges. Example: Prove Sice For all : 0 x2 x2! 2! x 2 x 2 =0 coverges for all real x.! 2 x 2 =0 is the Taylor! Series for f x = e x2. =0 =0 coverges for all real x, coverges for!! 2 all real x by the Direct Compariso Test. x 2
What about Negative Terms? Does the Direct Compariso Test fail if there are egative terms? Cosider the series =0 si 3π 2 si x =0 if x = 3π 2 :!! = + 2 6 + 24 20 Usig partial sums, it appears the series coverges: ; 0; 0.5; 0.333; 0.375; 0.367 But we do t have a test to prove the series coverges. Similar to the Ratio Test, what happes whe we look at the absolute value of each term: =0 si 3π 2! = + + 2 + 6 + 24 + 20 +! Usig partial sums, this series ALSO appears to coverges: ; 2; 2.5; 2.667; 2.708; 2.77 Sice the ew series is less tha the origial, if we ca prove the ew series coverges, the the origial series must coverge.
Absolute Covergece Test If Example: Prove = a coverges, the a = coverges. Ulike the Compariso Test, this test does ot require the terms to be positive. Ivestigate si x Sice! si x! si x! si x =0 coverges for all x. Notice: si x!!! =0 coverges by the Ratio Test:! coverges for all real x by the Direct Compariso Test, =0 coverges for all real x by the Absolute Covergece Test. lim a a lim! lim!!! lim 0
Defiitio of Absolute Covergece If = a coverges, the = a coverges absolutely. Example: The Alteratig Harmoic Series coverges but it does ot coverge absolutely: = = ( ) = 2 + 3 4 + ( ) = = = l 2 = + 2 + 3 + 4 + This is the diverget Harmoic Series.
Note about Absolute Covergece If = a coverges, the = a coverges absolutely. The Ratio Test uses a absolute value: a + a. Thus, every test that coverges by the Ratio Test also coverges absolutely: = 4( 0.5) coverges absolutely because 4(0.5) = coverges by by the ratio test The Ratio Test is icredibly strog