Lesson 22: Integrals and Series restart; with(Student[Calculus1]): with(IntegrationTools):

We were looking at this improper integral, which converges due to rapid oscillation. J := Int(x*cos(x^3), x=0..infinity); Maple's symbolic value for this integral was Jtrue := value(J); We split the interval into two parts, did a change of variables on the infinite part, and then some integrations by parts. with(IntegrationTools); S:=Split(J,1); J1:= op(1,S); J2:= op(2,S); V1:= evalf(ApproximateInt(x*cos(x^3),x=0..1,method=simpson,partition=50)); J3:= Change(J2,x=t^(1/3)); J4:= Parts(J3,t^(-1/3)); J5:= Parts(J4,t^(-4/3)); for k from 0 to 7 do J5 := Parts(J5,t^(-k-7/3)) end do; J6 := Change(J5,t=1/u); This time the integrand has four continuous derivatives, so Simpson ought to work. diff(u^(25/3)*cos(1/u),u$4); Again, we can get around the fact that the integrand is undefined at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjBGJ0ZARkBGK0ZARitGQA== by using piecewise. g:= piecewise(u=0,0,cos(1/u)*u^(25/3)); A50:= evalf(ApproximateInt(g,u=0..1,method=simpson,partition=50)); S50:= evalf(V1 + eval(J6,Int(cos(1/u)*u^(25/3),u=0..1)=A50)); A100:= evalf(ApproximateInt(g,u=0 .. 1,method=simpson,partition=100)); S100:= evalf(V1 + eval(J6,Int(cos(1/u)*u^(25/3),u=0..1)=A100)); SR := (2^4*S100-S50)/(2^4-1); evalf(Jtrue-SR); Not very accurate. Let's calculate A100 and A200. I'll also redo V1 with partition = 100 and partition=200; also increase Digits to reduce roundoff error. Digits:= 15; V1100 := evalf(ApproximateInt(x*cos(x^3),x=0..1,method=simpson,partition=100)); V1200:= evalf(ApproximateInt(x*cos(x^3),x=0..1,method=simpson,partition=200)); A100:= evalf(ApproximateInt(g,u=0..1,method=simpson,partition=100)); S100:= evalf(V1100 + eval(J6,Int(cos(1/u)*u^(25/3),u=0..1)=A100)); A200:= evalf(ApproximateInt(g,u=0 .. 1,method=simpson,partition=200)); S200:= evalf(V1200 + eval(J6,Int(cos(1/u)*u^(25/3),u=0..1)=A200)); SR := (2^4*S200-S100)/(2^4-1); evalf(Jtrue-SR); evalf(J); evalf(Jtrue-J);

Maple has three different commands for producing a sum: add, sum and Sum. The simplest is add, which just adds a given finite number of terms. restart; add(t^j, j = 1..20); Then there is sum, which looks for a formula for the sum. This can be used for both finite and infinite sums, i.e. series. sum(t^j, j = 1..20); sum(t^j, j=1..n); add(t^j, j=1..n); sum(t^j, j=1..infinity); If Maple can't find a formula it just returns the sum unevaluated. sum(exp(-k^2), k=0..n); sum(exp(-k^2), k=0..infinity); For a finite sum, sum and add generally return the same result, but there can be problems with "premature evaluation". Consider the following Vector: V := <1,2,3,4,5>; You want to add LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn for n from 1 to 5. This works fine: add(V[n], n=1..5); But this doesn't: sum(V[n],n=1..5); The problem is that sum starts out, like nearly all Maple procedures, by evaluating its inputs. In this case the first one is V[n], so it tries to evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGPQ==. But n is a symbolic variable. There is no such thing (to Maple) as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, just 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 it causes an error. This problem doesn't occur with add, because that has special evaluation rules so it only evaluates its first input with integer values substituted for the index variable. And finally there is the inert form Sum. Sum(t^j, j=1..20); You can use value to turn Sum into sum: value(%); To get numerical values, you can use evalf. This can deal with an infinite Sum (or a sum that Maple can't find a formula for) using numerical methods. evalf(Sum(1/k^2,k=1..infinity)); sum(1/k^2, k=1..infinity); evalf(%); sum(exp(-k^2),k=0..infinity); evalf(%); It's not always successful, though. evalf(Sum(1/(k^2+cos(k)),k=0..infinity));

Consider an infinite series 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 The n'th partial sum of the series is 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 The main theoretical question about a series is whether or not it converges. The sum of an infinite series is, by definition, the limit of the partial sums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, i.e. 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 That limit must exist (as a finite number) in order for the sum to exist. Otherwise, we say the series diverges. In a case where the partial sums go to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZGLUYsNi5RIn5GJ0YvRjJGNUY4RjpGPEY+RkBGQkZERkdGL0Yywe might write 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, but strictly speaking we should say this series diverges to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZGLUYsNi5RIi5GJ0YvRjJGNUY4RjpGPEY+RkBGQkZERkdGL0Yy Maple can sometimes figure this out, but not always. For a series that diverges to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, it might give a result of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: sum(1/k,k=1..infinity); But not always: sum(1/(k+ln(k)),k=1..infinity); A series that diverges, but not to +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRKCZpbmZpbjtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, will likely just return unevaluated: sum((-2)^k,k=0..infinity); But that's also what you get for a convergent series where Maple just doesn't know a formula for it. And evalf sometimes returns a finite answer for divergent series. evalf(%); There is an environment variable _EnvFormal that is supposed to make sum try harder to detect divergent series when it's set to false. It isn't always reliable, though. I'll do a restart so Maple won't remember previous results it got without _EnvFormal set to false. restart; _EnvFormal:=false; sum(1/(k+ln(k)),k=1..infinity); evalf(Sum((-2)^n,n=0..infinity)); That answer of 1/3 wasn't complete nonsense. It is actually the continuation of a correct formula outside the region where it is correct. Sum(a^n, n=0..infinity) = sum(a^n, n=0..infinity); That is true if 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. If we plug in 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: eval(%,a=-2); If _EnvFormal is set to true, Maple does even more of this sort of thing. restart; _EnvFormal := true; sum((-2)^n, n=0..infinity); So how do we tell whether a series converges or diverges? There are a number of "convergence tests" that can be used: the integral test, comparison test, ratio test, root test and alternating series test are the ones that tend to be found in calculus texts. Maple can be used as a tool in applying any of them. For example, the ratio test says the following: If 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 has a limit L < 1 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==, then 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 converges. If it has a limit L > 1 (possibly LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=), then the series diverges. If L = 1 or there is no limit, the ratio test is inconclusive. Maple can find limits with the limit command. For example, let's test the series 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 x:= k -> 3^(k^2)/k!; ratio := x(k+1)/x(k); limit(ratio, k=infinity); So this one diverges. It's not hard to see why. simplify(ratio); As LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiOUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI= grows faster than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=, so the ratio goes to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Next, I'll try 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 x := k -> k^k/(k!)^2; ratio := x(k+1)/x(k); limit(ratio,k=infinity); ratio := simplify(ratio); This one converges. What is the sum? S:=sum(x(k),k=1..infinity); Maple doesn't know a formula for it. We can get a numerical value using evalf. evalf(S);

Let's look at some partial sums of this series. seq( evalf(Sum(x(k), k=1 .. N)), N = 1 .. 10); This makes Maple's answer look plausible. The partial sums appear to be approaching a limit that is approximately evalf's answer. If we want to approximate the sum, we could use the partial sum for an appropriate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. What N should we use to get the sum S with an error of at most LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEYuRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=? We want to estimate 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, the "tail" of the series. The ratio test gives us a clue for this: the ratio was 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 Note that 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 goes to a limit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRidGLw== as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==. It is in fact less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRidGLw== for every LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: 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 = 1. So the ratio is less than 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. This means that for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= >= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= >= 10, we have the bound 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 And so for the tail of the series, we have an estimate: 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 By the way, the letter e is not anything special in Maple input. To get the constant e, you have to use exp(1). The right side is a geometric series. sum((exp(1)/11)^(k-N)*x(N),k=N+1 .. infinity); EstimatedR := simplify(%); We want this estimated R to be less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTEYuRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=. fsolve(EstimatedR = 10^(-10), N); So N = 19 should do it. evalf(eval(EstimatedR, N=19)); evalf(add(x(k),k=1..19)); evalf(S); evalf(S-Sum(x(k),k=1..19), 20); That was a series that converged very quickly. For series that converge slowly, approximating the sum can be more difficult.

sum Sum _EnvFormal

JSFH