Taylor Series Name:

Maple can be used to find, plot, and evaluate Taylor Series. The appropriate commands follow.

Find a Taylor Series of order 7 for the function 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 about x = 0. f:=x->sin(x); t:=taylor(f(x),x=0,8); The series must be converted to a polynomial in order to use it. p:=convert(t,polynom); Let's use the series to estimate 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 and compare it with the exact value of 0.50. First make the series into a function of x. p1:=unapply(p,x); evalf(p1(Pi/6)); Next, we will plot the function NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRK19jc3R5bGUyNjhGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYnRiwtRiQ2Ji1GLTY5USJmRihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1JI21vR0YlNjNRMCZBcHBseUZ1bmN0aW9uO0YoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0ZhcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlcvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkQtRiQ2JS1GZW82M1EiKEYoL0Zpb1EncHJlZml4RigvRlxwRjtGXXAvRmBwUS50aGlubWF0aHNwYWNlRigvRmNwRmVyL0ZlcEY7RmZwRmhwRltxRl5xRmBxRmJxRmRxRmZxRmhxRmpxLUYkNiMtRi02OVEieEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tRmVvNjNRIilGKC9GaW9RKHBvc3RmaXhGKEZjckZdcEZkci9GY3BRMnZlcnl0aGlubWF0aHNwYWNlRihGZ3JGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanFGLC1GZW82M1EiPUYoRmhvRltwRl1wL0ZgcFEvdGhpY2ttYXRoc3BhY2VGKC9GY3BGaHNGZHBGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanEtRiQ2Ji1GLTY5USRzaW5GKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuL0ZpblEnbm9ybWFsRihGW29GZG9GXHJGLEYsRiw3IzYjLy0lImZHNiMlInhHLSUkc2luR0ZndA== together with the series just found. plot({f(x),p},x=-2*Pi..2*Pi,y=-1.5..1.5,thickness=2); Observe that the series is close to the function. More terms in the series will give better accuracy. t1:=taylor(f(x),x=0,21); p2:=convert(t1,polynom); p3:=unapply(p2,x); evalf(p3(Pi/6)); plot({f(x),p,p2},x=-4*Pi..4*Pi,y=-1.5..1.5,thickness=2);

Exercise 1: Find Taylor Series of order 8, 12, and 16 for the function NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRK19jc3R5bGUyNzJGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYnRiwtRiQ2Ji1GLTY5USJmRihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1JI21vR0YlNjNRMCZBcHBseUZ1bmN0aW9uO0YoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0ZhcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlcvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkQtRiQ2JS1GZW82M1EiKEYoL0Zpb1EncHJlZml4RigvRlxwRjtGXXAvRmBwUS50aGlubWF0aHNwYWNlRigvRmNwRmVyL0ZlcEY7RmZwRmhwRltxRl5xRmBxRmJxRmRxRmZxRmhxRmpxLUYkNiMtRi02OVEieEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tRmVvNjNRIilGKC9GaW9RKHBvc3RmaXhGKEZjckZdcEZkci9GY3BRMnZlcnl0aGlubWF0aHNwYWNlRihGZ3JGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanFGLC1GZW82M1EiPUYoRmhvRltwRl1wL0ZgcFEvdGhpY2ttYXRoc3BhY2VGKC9GY3BGaHNGZHBGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanEtRiQ2Ji1GLTY5USRjb3NGKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuL0ZpblEnbm9ybWFsRihGW29GZG9GXHJGLEYsRiw3IzYjLy0lImZHNiMlInhHLSUkY29zR0ZndA== about x = 0, plot them together with the function, and use each series to approximate 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 and compare with the exact value. Exercise 2: Find Taylor Series of order 5, 10, and 15 for the function 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 about x = 0, plot them together with the function, and use each series to approximate e and compare with the exact value. Exercise 3: Use a Taylor series for the function 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 to approximate 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 and determine the least number of terms necessary so the the approximation is within 0.01 of the true value .

The Ratio Test can determine if a series is convergent. For a series 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 , if 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 , then the series converges if L < 1, diverges if L > 1, and can do either if L = 1. Once a series is determined to converge, the radius of convergence can be found using the inequality 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. Example 1: Determine if the Taylor series for 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 and find the interval of convergence. restart; Since the series for 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 contains only even powers of x, the following commands must be used to divide a term by the preceding non-zero one. t1:=convert(taylor(cos(x),x=0,505),polynom): t2:=convert(taylor(cos(x),x=0,503),polynom): t3:=convert(taylor(cos(x),x=0,501),polynom): evalf(t1-t2); evalf(t2-t3); (t1-t2)/(t2-t3); The denominator of the above fraction depends on n, so as n becomes large, the fraction approaches zero. Since zero is less than 1, the series converges by the ratio test. The limit is zero for any value of x. Therefore the series converges for all x. Example 2: Determine the interval of convergence of the Taylor series for 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 . restart: t1:=convert(taylor(1/(1-x),x=0,505),polynom): t2:=convert(taylor(1/(1-x),x=0,504),polynom): t3:=convert(taylor(1/(1-x),x=0,503),polynom): evalf(t1-t2); evalf(t2-t3); (t1-t2)/(t2-t3); solve(abs(x)<1); This give the open interval -1 < x < 1. The endpoints must be tested separately. t4:=convert(taylor(1/(1-x),x=0,10),polynom); If 1 is substituted for x in the series above, it keeps growing and therefore doesn't converge. If -1 is substituted, the sum alternates and doesn't converge. The interval of convergence is thus the open interval stated above.

Exercise 1: Determine the interval of convergence for the Taylor series of 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 . Exercise 2: Determine the interval of convergence for the Taylor series of 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. Exercise 3: Use a Taylor series of order 20 to approximate the integral 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 and compare it with the answer given by the int command.