ONE-PAGE REVIEW. (x c) n is called the Taylor Series. MATH 1910 Recitation November 22, (Power Series) 11.7 (Taylor Series) and c
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1 ONE-PAGE REVIEW 6 (Power Series 7 (Taylor Series MATH 90 Recitatio November 22, 206 ( A ifiite series of the form F(x is called the ceter (2 (2 The radius of covergece a (x c is called a power series ( ad c (3 of F(x a (x c is a costat R such that F(x coverges absolutely for x c < R ad diverges for x c > R If F(x coverges for all x, the R ( (3 To determie R, use the ratio test ( x x (6 ( The powerseries T(x c 0, this is called a, with R ( (7 f ( (c (x c is called the Taylor Series Maclauri series (9 (6 + x + x2 2! + x3 3! + x (0 ex (7 ( + x a + ( a x for x <, where ( a (8 for f (x If a(a (a 2 (a + ( Hadout format by Drew Zemke
2 SOLUTIONS 6 (Power Series 7 (Taylor Series MATH 90 Recitatio November 22, 206 ( Show that all three of the followig power series have the same radius of covergece, but differet behavior at the edpoits (x (a 9 Use the ratio test to determie the radius of covergece ρ lim a + a lim So this series coverges if 9 <, ad has radius of covergece R 9 But ow we eed to check the edpoits, which are x ad x x : x : ( 9 ( 9 diverges ( diverges So the iterval of covergece is (, (b (x 9 Use the ratio test to determie the radius of covergece ρ lim a + a lim + 9 ( So this series coverges if 9 <, ad has radius of covergece R 9 But ow we eed to check the edpoits, which are x ad x x : x : ( 9 ( 9 ( diverges coverges The iterval of covergece is [, (c (x 2 9 Hadout format by Drew Zemke
3 Use the ratio test to determie the radius of covergece ρ lim a + a lim ( So this series coverges if 9 <, ad has radius of covergece R 9 But ow we eed to check the edpoits, which are x ad x x : x : ( 2 9 ( ( 2 coverges coverges The iterval of covergece is [, ] (2 Use the geometric series formula to expad the fuctio ceter c 0 ad determie radius of covergece The formula for the geometric series implies that x for x < Replace x by 3x i that formula to get x + 3x + 3x ( 3x ( 3 x i a power series with This formula is valid for 3x <, or x < /3 So the radius of covergece is R 3 (3 Write out the first four terms of the Taylor series f (x cetered at c 3 if f (3, f (3 2, f (3 2, f (3 3 f (x f (3 + f (3(x 3 + f (3 2! (x f (3 (x ! + 2(x ! (x ! (x (x 3 + 6(x (x 33 + ( Fid the Taylor series of the followig fuctios ad determie the radius of covergece Hadout format by Drew Zemke
4 (a f (x si(2x, cetered at x 0 si(x si(2x ( x 2+ (2 +! ( (2x2+ (2 +! ( 2 2+ x 2+ (2 +! Sice the formula for si(x is valid for all x, the formula for si(2x is also valid for all x (b f (x e x, cetered at x 0 e x e x x (x x Sice the formula for e x is valid for all x, so is the formula for e x (c f (x x 2 e x2, cetered at x 0 (x e x2 2 x 2+2 ( x 2 e x2 x 2 x 2+2 x 2+2 Sice the formula for x 2 is valid for all x, so is the formula for x 2 e x2 (d f (x, cetered at c 3x 2 Rewrite the fuctio as follows: 3x 2 + 3(x + Now use the geometric series formula, valid for x < 3x 2 ( 3(x + This formula is ow valid for is 3 3(x+ 3(x+ 3 (x + 3 (x + + <, or x + < 3 So the radius of covergece Hadout format by Drew Zemke
5 (e f (x ( + x /3, cetered at c 0 Use the biomial series formula with a 3 ( + x 3 + ( 3 x The radius of covergece is, sice the formula is valid for x < (f f (x x, cetered at c First rewrite the fuctio ( x + (x + x 2 + x Now fid the MacLauri series of + u by settig a 2 i the biomial series formula ( 2 ( + u 2 + u + u This is valid for u < Now replace u by x to get + x ( 2 ( x + + This is valid for x The fial aswer is: ( 2 (x < or x < So the radius of covergece is x 2 + ( 2 2 (x If you re willig to do a lot of simplifyig, you ca evetually get to: x 2 + ( (2 2! 2 2 ( 2 (x Hadout format by Drew Zemke
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