Lesson 24: Taylor series restart;

As we saw last time, Maple has the taylor command to find a given number of terms of a Taylor series of an expression. taylor((exp(x)-1-x)/x^2, x); But what if instead of the first few terms of the series, you want a formula for the whole series? This isn't always going to work, but sometimes it will: convert((exp(x)-1-x)/x^2, FormalPowerSeries, x); value(%); If you want a series about some other point, say LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJhRidGNkY5RkBGK0ZARitGQA==: convert(exp(x), FormalPowerSeries, x=a); convert(exp(x+x^2), FormalPowerSeries,x); Maple doesn't know a formula for the coefficients of the Maclaurin series of 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. Well, there isn't a very nice formula for them, as far as I know. taylor(exp(x+x^2),x,10);

Let's look at the Maclaurin series for the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy and its convergence to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy. It's convenient to define a function that will calculate the n'th degree Maclaurin polynomial at a given point. P:= (n,t) -> eval( convert(taylor(exp(x), x=0, n+1), polynom), x=t); P(4,t); P(4,2); plot([ exp(x), seq(P(n, x), n=1 .. 10)], x=-6 .. 2, y=-2 .. 8); Especially for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk=, it's more informative to look at the difference between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy and the Maclaurin polynomial. plot([seq(exp(x)-P(n, x), n=1 .. 12)], x = -6 .. 6, y = -3 .. 3); An animation is another possibility. I'm not using animate here because premature evaluation would cause trouble. with(plots): display([seq(plot(exp(x)-P(n,x),x=-7..7,y=-3..3, title=('n'=n)), n=1..16)], insequence=true); It's almost (but not quite) true that the curves for x > 0 march off to the right at a constant rate of 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 per step. display([seq(plot(exp(x+n*exp(-1)) - P(n, x+n*exp(-1)), x=-1 .. 1.5, y = -1 .. 2,title=('n'=n)),n=1..50)],insequence=true); We'd get a very different picture for e.g. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSdhcmN0YW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3. Here's the Taylor series: convert(arctan(x),FormalPowerSeries,x); P := (n,t) -> subs(x=t,convert(taylor(arctan(x), x=0, n+1), polynom)); P(7,t); P(8,t); plot([seq(arctan(x)-P(2*n,x), n=1..12)], x=-2 .. 2, y = -3 .. 3); display([seq(plot(arctan(x)-P(2*n,x),x = 0.7 .. 1.3, y = -3 .. 3,title=('Order'=2*n)), n = 1..30)], insequence=true); The difference here is that the radius of convergence for arctan is 1, while for exp it is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Outside the interval [-1,1], the series for arctan is useless for approximating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSdhcmN0YW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3.

Various operations can be done to obtain new series from old series: the basic operations of arithmetic, as well as substitution, differentiation and integration.

Starting with series any good Math 101 student should know, obtain the degree 10 Maclaurin polynomial for 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. For example, the series for 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 is a geometric series. s1 := 1/(1-t) = convert(taylor(1/(1-t), t=0, 10),polynom); Of course that's not literally true, it's just the first part of the series. But for some of the manipulations to work, I want a polynomial rather than a series. Integrate this term-by-term and you have the series for 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. Note that the constants of integration are the same, because both sides are 0 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. s2 := int(lhs(s1),t) = int(rhs(s1),t); Change signs and substitute 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 and you have the series for 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. s3 := eval(-s2, t=-x^2); That's way more terms than we need, we only want a degree-10 polynomial. s4 := lhs(s3) = convert(taylor(rhs(s3),x,11),polynom); That's one of the factors. Now for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJGNvc0YnRi9GMi1GNjYkLUYjNiQtRiw2JVEieEYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMkYyRjJGMg==. The Maclaurin series for cos(x) and sin(t) are also "known". convert(cos(x),FormalPowerSeries,x); convert(sin(x),FormalPowerSeries,x); s5 := cos(x) = convert(taylor(cos(x),x,11),polynom); s6 := sin(x) = convert(taylor(sin(x),x,11),polynom); Now I want 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, which is approximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSNzNUYnL0Y1USV0cnVlRicvRjhRJ2l0YWxpY0YnRjdGN0Y3RitGNw==. But it would be wrong to use the Taylor series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is small, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 is near 1, not near 0. So we want a series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=. A trigonometric identity relates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== to sin and cos of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=: trigident:= eval(sin(1+s)=expand(sin(1+s)),s=t-1); s7 := eval(s5,x=t-1); s8 := eval(s6,x=t-1); s9 := eval(trigident,{s7,s8}); s10 := sin(t) = convert(taylor(rhs(s9),t=1,11),polynom); s11 := eval(lhs(s10),t=cos(x)) = eval(rhs(s10),t = rhs(s5)); s12 := lhs(%) = convert(taylor(rhs(%),x,11),polynom); And finally: lhs(s4)*lhs(s12)=convert(taylor(rhs(s4)*rhs(s12),x,11),polynom); Of course, we could have used one "taylor" command, this was just to see how it could be done. taylor(ln(1+x^2)*sin(cos(x)),x,11); taylor(rhs(%%)-%, x, 11);

Find the Taylor series for 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 about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMCwxMjhdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGMkY8RjJGPA== up to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMCwxMjhdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI2RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjVGQQ== term, if 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 satisfies the equation 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 with 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. eq:= (1+x)*exp(y)-y^2*exp(x)=1+ x^2*y; Check that it works with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSJ+RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVLUZBNi1RIj1GJ0Y+RkRGR0ZJRktGTUZPRlEvRlRRLDAuMjc3Nzc3OGVtRicvRldGZm5GQEY6Rj4=. eval(eq,{x=0,y=0}); So we want the first 6 terms to look like this: yseries := add(a[n]*x^n,n=1..6); Substitute this in to the difference of the two sides of the equation. eval(lhs(eq)-rhs(eq), y=yseries); taylor(%, x, 7); Now the coefficients of each power of x should match. equations:= {seq(coeff(%,x,n),n=1..6)}; solve(equations); So here is our answer. answer:= eval( yseries, %); I'll check this graphically. with(plots): P0:= implicitplot(eq,x=-1..1,y=-1..2,colour=blue): for j from 1 to 6 do frame[j]:= display([P0,plot(convert(taylor(answer,x,j+1),polynom),x=-1..1)],title=('Degree'=j)) end do: display([seq(frame[j],j=1..6)],insequence=true,view=[-1..1,-1..2]); Here's a look at the curve on a larger scale. implicitplot(eq,x=-10..10,y=-10..10,gridrefine=3); Rather than doing this with solve, you can use a version of Newton's method. f:= unapply(lhs(eq)-rhs(eq),(x,y)); newt:= (y,n) -> convert(normal(taylor(y-f(x,y)/D[2](f)(x,y), x, n)),polynom); y1 := newt(0,2); y2 := newt(y1,4); normal(taylor(y2 - answer,x,4)); How does this work? If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUklbXN1YkdGJDYlLUYsNiVRInlGJ0YvRjItRiM2JS1GLDYlUSJrRidGL0YyRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkFGQS1GPjYtUSJ+RidGQUZDL0ZHRkVGSEZKRkxGTkZQRlIvRlZGVC1GPjYtUSI9RidGQUZDRmNvRkhGSkZMRk5GUC9GU1EsMC4yNzc3Nzc4ZW1GJy9GVkZpb0Zgby1GLDYlUSJPRicvRjBGRUZBLUY2NiQtRiM2JC1JJW1zdXBHRiQ2JUY6RmhuLyUxc3VwZXJzY3JpcHRzaGlmdEdGX29GQUZBRkE= and 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 has a nonzero limit as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUkjbW5HRiQ2JFEiMEYnRjlGOQ==, then I claim 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 where 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 In fact 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 and 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 In other words, once you get an approximation that works to a certain order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JS1GLDYlUSJrRidGQEZCRkBGQi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMkYyLUYsNiNRIUYnRjI=, each application of Newton's method will at least double the order of approximation.

convert(..., FormalPowerSeries, ...)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=