Lesson 23: Approximating the sum of a series restart;

Let's look at one that evalf couldn't do until recently (it couldn't do it in Maple 11, but can in Maple 12,13 and 14). S := Sum(1/(k^2+ln(k)),k=1..infinity); evalf(S); x:= k -> 1/(k^2 + ln(k)); Does the ratio test work on this one? ratio := x(k+1)/x(k); limit(ratio,k=infinity); No, it's inconclusive. What does work is a comparison test. If 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 converges and 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 for all k >= N, then 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 converges. Moreover, we have an estimate on the tails: 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. In this case we can take 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, as this is a "well-known" convergent series: sum(k^(-2),k=1..infinity); And it's clear that 0 < 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 for all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUS8mR3JlYXRlckVxdWFsO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZTLUkjbW5HRiQ2JFEiMUYnRjlGOQ==. What's the estimate on the tail? EstimatedR := sum(k^(-2),k=N+1..infinity); fsolve(EstimatedR = 10^(-10)); Oops. I don't really want to add 10 billion terms to get a good value for the sum of this series. Fortunately there are other ways to do it. One way is based on the integral test. The idea is to get both good upper and lower bounds for the tail of the series, and approximate our sum using these. Suppose x(t) is a positive decreasing function for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUS8mR3JlYXRlckVxdWFsO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZTLUkjbW5HRiQ2JFEiMUYnRjlGOQ==. Then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNigtSSttdW5kZXJvdmVyR0YkNictSSNtb0dGJDYvUSYmU3VtO0YnLyUrZm9yZWdyb3VuZEdRLlsxNDQsMTQ0LDE0NF1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEia0YnLyUnaXRhbGljR0ZKL0Y+USdpdGFsaWNGJy1GNzYtUSI9RidGPUZDRkYvRklGRUZLL0ZORkUvRlBGRUZRL0ZUUSwwLjI3Nzc3NzhlbUYnL0ZXRmNvLUkjbW5HRiQ2JFEiMUYnRj1GPS1GLDYlUSgmIzg3MzQ7RidGaG5Gam5GUS8lLGFjY2VudHVuZGVyR0ZFRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJDUuMEYnLyUmZGVwdGhHRmNwLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYjNiYtRiw2JVEieEYnRmhuRmpuLUY3Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj1GQ0ZGRl9vRktGYG9GYW9GUUZTL0ZXRlUtSShtZmVuY2VkR0YkNiQtRiM2JEZlbkY9Rj1GPUYrRj1GK0Y9RitGPQ== converges if and only if 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 converges. Moreover, we have an estimate on the tails: 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 This can be seen in the following pictures (where now we're approximating a sum by integrals instead of approximating an integral by sums) : with(Student[Calculus1]): ApproximateInt(x(t),t=5..15,method=left,partition=10,output=plot); This shows that if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is decreasing 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 ApproximateInt(x(t),t=4..14,method=right,partition=10,output=plot); This shows that if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is decreasing 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 The idea now is not just to make the tail small, but to use good estimates on both sides. If 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 is between A and B, we can use (A+B)/2 as an approximation for it, with error at most (B-A)/2. In this case A = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ki1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GIzYmLUYsNiVRIk5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GPFEnaXRhbGljRictRjU2LVEiK0YnRjtGQUZERkZGSEZKRkxGTi9GUVEsMC4yMjIyMjIyZW1GJy9GVEZdby1JI21uR0YkNiRRIjFGJ0Y7RjstRiw2JVEoJmluZmluO0YnRlpGZ24vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYmLUYsNiVRInhGJ0ZaRmduLUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjtGQUZERkZGSEZKRkxGTkZQRlMtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGWkZnbkY7RjtGO0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmFxLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY1Ni9RMCZEaWZmZXJlbnRpYWxEO0YnRjhGO0Y+RkFGREZGRkhGSkZMRk5GUEZTRmlwRjtGK0Y7 and B = 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 so B-A = 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. int(x(t),t=N..N+1); Well, that's less than what we'd get without the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSNsbkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJy9GNVEldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3RjdGN0YrRjc=. int(1/t^2,t=N..N+1); Oops, better assume N>0. % assuming N>0; solve(%/2 = 10^(-10),N); evalf(%[2]); OK, N = 70711 should do. N := 70711; A:= evalf(Int(x(k),k=N+1..infinity)); B:= evalf(Int(x(k),k=N..infinity)); The next command would take a rather long time # Sapprox:= add(evalf(x(k)),k=1..N)+(A+B)/2; The result would be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoU2FwcHJveEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEsMS41ODQ2MjA3MTBGJ0Y5Rjk=. Much faster is to use hardware floats using evalhf. This also should improve accuracy since it's using about 15 Digits instead of 10. Sapprox := evalhf(add(x(k),k=1..N)+(A+B)/2); Digits:= 15; SMaple:= evalf(S); Sapprox - SMaple;

70711 is a lot of terms to calculate. Let's see if we can do better. The bounds we used were essentially comparing 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 to the values of x at the two endpoints: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJuRidGNEY3Rj5GPkY+RitGPg== and 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. We're using an integral to approximate a sum, but we can relate this to what we've done before in approximating integrals by sums. And using the value of x at the left or right endpoint is a very crude way to approximate the integral. For an improvement, we might try the Midpoint Rule: approximate 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 by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJuRidGNEY3Rj5GPkY+RitGPg==. What's the error in this approximation? (I'll change from x to f temporarily, because I want something valid for arbitrary functions). L:= h -> Int(f(t),t=n-h .. n + h) - 2*h*f(n); L(0),D(L)(0); (D@@2)(L)(h); If 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImhGJ0Y0RjdGPkYrRj4=, then 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGOC1GLDYlUSJiRidGU0ZWRjVGK0Y1, and integrating twice we get 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 < 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. In particular, take 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 to get: 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIzI0RidGOkY6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjo= i.e. 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 < 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 where 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for 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 < 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. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnZicnKHQpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== is a decreasing function of t, we can take 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 and 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. Is that true for our function x? normal(diff(x(t),t$3)); Yes, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRidGNUY5 is decreasing, at least if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjFGJ0Y5Rjk= (note that 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, so 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). So we have 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 < 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 Adding this up for n from N+1 to infinity: 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 < 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 To avoid having to sum the second derivatives, use integral estimates: since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkZicnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== is decreasing and 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 < 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 while 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 > 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. Thus 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 < 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 This estimate is true whenever LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRidGNUY5 is decreasing on 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. We want to take N large enough so the difference between the two bounds is less than 2*10^(-10). fsolve((D(x)(t+3/2)-D(x)(t-1/2))/24=2*10^(-10),t=1..infinity); So N = 224 should do. N := 224; Here's the integral: J:= evalf(Int(x(t),t=N+1/2 .. infinity)); Here are the two bounds. lowerbd:= evalf(add(x(n),n=1..N) + J + D(x)(N-1/2)/24); upperbd:= evalf(add(x(n),n=1..N) + J + D(x)(N+3/2)/24); Their average is our estimate. S2:= (lowerbd+upperbd)/2; This agrees quite well with the previous answer we got: Sapprox; S2-SMaple;

Probably the most important series are power series. These have very important applications in many areas of mathematics. A power series about the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImNGJ0Y0RjdGPkYrRj4= is a series of the form 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 . A power series has a radius of convergence LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (possibly 0 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=): the series converges when 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 and diverges when 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 > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The Taylor series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== about the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImNGJ0Y0RjdGPkYrRj4= is the power series 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 In the case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= this is also called the Maclaurin series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg==. The function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== is said to be analytic at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImNGJ0Y0RjdGPkYrRj4= if its Taylor series about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJjRidGL0YyRjk= converges to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in some open interval around LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImNGJ0Y0RjdGPkYrRj4=. Most of the functions we're familiar with are analytic, except perhaps at some exceptional points. The partial sums of the Taylor (or Maclaurin) series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiw2JVEieEYnRjRGN0Y+Rj5GK0Y+ are called the Taylor (or Maclaurin) polynomials.

Maple has the taylor command to find a given number of terms of the Taylor series of an expression. You could also use series, which is somewhat more general (we saw it in Lesson 21). S:=taylor(exp(x), x=2, 5); This is part of the Taylor series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GPkYrRj4=. The third argument (5 in this case) controls how many terms you get in the result. Usually, taylor(f(x),x=c, n) will give you the terms up to 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, and the rest are indicated by 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, but not always. Here you get too few terms: A:= taylor((exp(x)-1-x)/x^2, x=0, 5); Here you get a bit more than you asked for: B:= taylor(x^3*sin(x^3), x=0, 8); What actually happens is that n represents the maximum number of terms Maple uses in computing the Taylor series, and this is not necessarily the number of terms in the final result. For example, in the case of (exp(x)-1-x)/x^2, here's what happens. Start with taylor(exp(x), x=0, 5); Subtract LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUScmcGx1cztGJ0Y1LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk0tRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1RitGNQ==. taylor(% - (1+x), x=0, 5); Then divide by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4=. taylor(%/x^2, x=0, 5); The bottom line is, if you get fewer terms than you need, you should start with a larger n. taylor((exp(x)-1-x)/x^2, x=0, 7); By the way, if you want a Maclaurin series, you don't need the "=0": taylor((exp(x)-1-x)/x^2,x,7); The result of taylor looks like an ordinary polynomial with an extra term 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, but it's really something a bit different: a special "series" data structure. The whattype function tells you what type of data structure something is. whattype(S); lprint(S); In many cases, if you want to do something with a series, you'll need the Taylor polynomial rather than this "series" structure. You can't just subtract the 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, you must use convert(..., polynom): convert(S, polynom); whattype(%);

taylor whattype convert(..., polynom)

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