Lesson 27: Asymptotic Series restart;

We were looking at the following integral as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCYjODU5NDtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSgmIzg3MzQ7RidGNkY5RkBGK0ZARitGQA==. J := Int(exp(-t)/t, t=x..infinity); Jv:= value(J); asympt(%,x,10); JSFH Our series was 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 which is an asymptotic series that diverges for all x. Let's see how well the partial sums of our asymptotic series do at approximating the original integral J. for count from 1 to 30 do PS[count]:= exp(-x)*add((-1)^k*k!/x^(1+k),k=0..count-1) end do: The more terms we have in the series, the better it is at approximating J when x is large. But how large is "large" depends on n. For any particular x, the approximations typically get better for a while, but then eventually get worse, because the series diverges. Here are the errors for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjVGJ0Y5Rjk=, for example. Errors5:=[seq([n, evalf(eval(Jv-PS[n],x=5))], n=1..30)]; What's the smallest error (in absolute value)? First get the absolute values. map(t -> abs(t[2]), Errors5); Then take the minimum using min. min(%); Which entry had this (or -this)? select(has,Errors5,{%,-%}); Let's try some animations. In the n'th frame, I'll plot J - PS[n] from x = 1 to 10. with(plots): display([seq(plot(Jv - PS[n], x=1..10,title=('n'=n)), n=1..30)], insequence=true,view=[1..10,-1..1]); That's maybe a pessimistic view: it shows really bad approximations for larger and larger x as n increases. It doesn't show the good approximations very well. A better idea, showing both the good and bad, comes from a logarithmic plot. The logplot command in the plots package plots an expression (which should be positive) using a logarithmic scale for the y axis. Here the approximation is better if the value of the log is more negative. display([seq(logplot(abs(Jv - PS[n]), x=1..20, title=('n'=n)), n=1..30)],insequence=true,view=[1..20,10^(-19)..1]);

Here is an example that actually came up a while ago in a question I was asked by a friend. The problem arose from research having to do with approximation of functions, and turns out to require solving the equation 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. Note: cosh is the hyperbolic cosine function cosh(x) = convert(cosh(x),exp); eq:= cos(x)*rhs(%)=-1; solve(eq, x); It seems the equation can't be solved in closed form. But it's easy to see that there will be a solution in each interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUScmIzk2MDtGJy9GMEY9RjlGNS1GNjYtUSI8RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORldGNS1GLDYlUSJ4RidGL0YyRjVGU0Y1LUkobWZlbmNlZEdGJDYkLUYjNiZGKy1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl9vLUkjbW5HRiQ2JFEiMUYnRjlGOUY5RjVGT0Y5 for integers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, because the left side will be positive at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSJuRidGL0YyRjUtRiw2JVEnJiM5NjA7RicvRjBGPUY5RjVGOQ==for even n and less than -1 for odd n. plot([cos(x)*cosh(x),-1],x=0..3*Pi,-5..5); In fact there is exactly one solution in each of these intervals: call it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEibkYnRjdGOi9GO1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZCRitGQg==. I want to find good approximations for these solutions. Of course fsolve can do it for any particular n, but I want a general formula. We should expect LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEibkYnRjdGOi9GO1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZCRitGQg== to be near 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, where the left side of the equation is 0. So I'll write 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. eq1:= expand(eval(eq,x=(n+1/2)*Pi+t)); Doesn't Maple know that 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? It does if we tell it that n is an integer. eq1:= eq1 assuming n::integer; Let's make a new variable 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. eq2:= eval(eq1,exp(Pi*n)=u); How does LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= behave when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is large? asympt(RootOf(eq2,t),u,15); S:= map(simplify,%) assuming n::integer; evalf(%); It looks like this should converge for approximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjFGJ0Y5Rjk=, thus maybe not for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjBGJ0ZARkBGK0ZARitGQA== but certainly for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUS8mR3JlYXRlckVxdWFsO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZTRjUtSSNtbkdGJDYkUSIxRidGOUY5. Sn := eval(S, u = exp(Pi*n)); Even for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjFGJ0ZARkBGK0ZARitGQA==, the first four terms are already quite accurate. S4:= convert(asympt(S,u,5),polynom); Digits:= 15; x1:= fsolve(eq, x=Pi .. 2*Pi); x1app:= evalf(3/2*Pi+eval(S4,{u=exp(Pi),n=1})); x1-x1app; JSFH

restart; Suppose we have some nice function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA, and we want LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA so that 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This is sometimes called an anti-difference of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA (by analogy with anti-derivative). For example, if we're just concerned with integers, we could take 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. Our main application is going to be in looking at sums like that. Note that, just like anti-derivatives, anti-differences are not unique, because you can always add a constant. In fact, you can add any function that's periodic with period 1. On a purely formal level, we can proceed as follows. Using Taylor series, 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 (for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjBGJ0ZARkBGK0ZARitGQA== term, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiQtSSVtc3VwR0YkNiUtRiw2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnRjxGPEY8RjwvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZJRjxGK0Y8RitGPA== is the identity operator, which we'll write as 1). So we can say we want to solve 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. What this really means is: expand 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 as a Maclaurin series in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and replace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEifkYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYvNiVRIm5GJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y+ by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRi82JVEibkYnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU=. Well, we might write 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Now 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 has a series in powers of d: series(1/(exp(d)-1),d,10); The differentiation operator D doesn't really have an inverse, but we can say integration is the closest thing there is to one. So we might try something like this as an approximation to an antidifference of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA (neglecting terms involving the 8th and higher derivatives): G:= int(f(x),x) - 1/2*f(x) + 1/12*D(f)(x) - 1/720*(D@@3)(f)(x) + 1/30240*(D@@5)(f)(x) - 1/1209600*(D@@7)(f)(x); Amazingly enough, this works. For example, try it on a polynomial of degree 7 (where those higher derivatives are 0) and it should work perfectly. f7:= unapply(add(a[j]*x^j,j=0..7),x); G7:= unapply(eval(G,f=f7),x); normal(G7(t+1)-G7(t)) = f7(t); The coefficients in the Maclaurin series for 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 turn out to be expressed in terms of Bernoulli numbers: FunctionAdvisor(identities,bernoulli(n))[1,2]; so our formula can be written as 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 This is usually a divergent series, so what we really have is (for fixed N) 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 where the remainder 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 depends on 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: if 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 for 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, then 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 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCYjODU5NDtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSgmIzg3MzQ7RidGNkY5RkBGK0ZARitGQA==. Here are the first few Bernoulli numbers: seq(b(n)=bernoulli(n),n=0..20); Except for n=1, the odd ones are 0. That's because 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 is an even function. q:= d -> d/(exp(d)-1)+d/2; simplify(q(d)-q(-d)); Maple has a command for the Euler-Maclaurin formula: eulermac. eulermac(f(x), x, 7); Now, what can we do with the Euler-Maclaurin formula?

The series 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 converges, but slowly. In Lesson 24 we developed ways of getting estimates for such series using upper and lower bounds. The best we had was to estimate the tail of the series (when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRidGNS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLRk9GTUY5 is decreasing) as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNidGKy1GIzYqLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2L1EmJmludDtGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUYjNiYtRiw2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnL0ZAUSdpdGFsaWNGJy1GOTYtUSIrRidGP0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjIyMjIyMjJlbUYnL0ZYRmFvLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnRj9GPy1GIzYkLUZpbzYkUSIyRidGP0Y/LyUubGluZXRoaWNrbmVzc0dGW3AvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXAvJSliZXZlbGxlZEdGR0Y/LUYsNiVRKCZpbmZpbjtGJ0ZobkZbby8lMXN1cGVyc2NyaXB0c2hpZnRHRmBwLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYmLUYsNiVRImZGJ0ZobkZbby1GOTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y/RkVGSEZKRkxGTkZQRlJGVEZXLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRmhuRltvRj9GP0Y/RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGZ3IvJSpsaW5lYnJlYWtHUSVhdXRvRictRjk2L1EwJkRpZmZlcmVudGlhbEQ7RidGPEY/RkJGRUZIRkpGTEZORlBGUkZURldGX3JGP0Zdby1GZG82KC1GIzYmRistRiM2Ji1GLDYlUSNmJ0YnRmhuRltvRmdxLUZbcjYkLUYjNiZGKy1GIzYmRmVuLUY5Ni1RKCZtaW51cztGJ0Y/RkVGSEZKRkxGTkZQRlJGYG9GYm9GY29GP0YrRj9GP0Y/RitGPy1GIzYkLUZpbzYkUSMyNEYnRj9GP0ZhcEZjcEZmcEZocEY/LUY5Ni1RIjxGJ0Y/RkVGSEZKRkxGTkZQRlIvRlVRLDAuMjc3Nzc3OGVtRicvRlhGXnUtRiM2KC1JK211bmRlcm92ZXJHRiQ2Jy1GOTYtUSYmU3VtO0YnRj9GRUZIL0ZLRmpuRkwvRk9Gam4vRlFGam5GUkZUL0ZYUSwwLjE2NjY2NjdlbUYnLUYjNictRiw2JVEibkYnRmhuRltvLUY5Ni1RIj1GJ0Y/RkVGSEZKRkxGTkZQRlJGXXVGX3UtRiM2JkZlbkZdb0Zob0Y/RitGP0ZqcEZSLyUsYWNjZW50dW5kZXJHRkdGKy1GY3I2JkZlci9GaXJRJDUuMEYnRltzRl1zLUYjNiZGZHFGZ3EtRltyNiQtRiM2JEZfdkY/Rj9GP0YrRj9GK0Y/RitGPw== < 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 With the Euler-Maclaurin formula, we can do better. Note that 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 is an antidifference of 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, since 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. So according to Euler-Maclaurin: T(x) = eulermac(-f(x),x); We're interested in cases where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== and its derivatives go to 0 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==, and so of course does LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJURicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg==. So the antiderivative in this formula should be the one that goes to 0 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==, namely 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. Thus we get an asymptotic formula for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJURicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg==: 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. I'll try this first with 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 (which is a function for which both the sum and the integral can be done in closed form): f1:= x -> 1/(x*(x+1)); Sum(f1(j),j=x..infinity) = sum(f1(j),j=x..infinity); Int(f1(t),t=x..infinity) = int(f1(t),t=x..infinity) assuming x > 0; Remainder:= sum(f1(j),j=x..infinity) - int(f1(t),t=x..infinity) + add(bernoulli(n)*(D@@(n-1))(f1)(x)/n!,n=1..10) assuming x > 0; asympt(%,x,15); There's also a form of eulermac that let's you specify that you want a formula for 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: you just say eulermac(f(j),j=a..b). Thus: eulermac(f1(t),t=x..infinity,10); We don't have very explicit bounds, but it's often the case that successive terms of the Euler-Maclaurin series (after the first few) have opposite signs, and typically the actual tail T(x) is between the Euler-Maclaurin sums for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= and 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. For example, in this one I claim that the following three values are in increasing order: L := [convert(eulermac(f1(t),t=x..infinity,8),polynom), sum(f1(t),t=x..infinity), convert(eulermac(f1(t),t=x..infinity,10),polynom)] assuming x > 0; I'm going to need to increase Digits somewhat, because roundoff error will be ugly. Digits:= 17: plot([(L[2]-L[1])*x^11,(L[3]-L[2])*x^13],x=1..10,colour=[red,blue]); To get the best possible approximation for our sum (with a fixed x) using Euler-Maclaurin series, we take more and more terms until the values stop getting closer together. I'll try it for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjJGJ0ZARkBGK0ZARitGQA==: we want the best Euler-Maclaurin approximation for 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 (the actual value is 1/2). eval([seq([2*n,eulermac(f1(t),t=x..infinity,2*n)],n=1..20)],{x=2,O=0}); L:=evalf(%); By the way, why didn't I just use eulermac(f1(t),t=2..infinity,2*n)? Because it wouldn't work right: eulermac needs the interval to involve a parameter that's supposed to go to infinity. eulermac(f1(t),t=2..infinity,2); [seq([j,L[j,2]-L[j-1,2]],j=2..20)]; map(t -> abs(t[2]),%); select(has,%%,{min(%),-min(%)}); The smallest absolute difference is at j=7, corresponding to L[6] and L[7] L[6],L[7]; So the true value should be somewhere between these. We might try the average of these. (L[6,2]+L[7,2])/2;

min logplot (in plots package) cosh bernoulli eulermac

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=