Hi @Pythagoras may i know how do you got the matrix above? Is it by guess?

They are just the coefficients in $A,B,\cdots$. Note that in $B$, they have a typo as well: It should read $B=y(1)-y(0)=y'(0)+\frac 1 2y''(0)+\cdots.$ The formulas for $A,B,C,\cdots$ follow from the Fundamental Theorem of Calculus.

i got it now lol

Thank you for your help !! So kind

drive.google.com/file/d/1sxTiXSXNuM6cWQ-_C31EZR2oKrgOeO_8/…

Hi Sir, Shouldn't B which is y(1)-y(0) be y(1)+y(0) as stated in Wikipedia. I shared the link above to check it out.

I am confused on that part only, cuz it must match the exact formula stated in Wiki :))

A is the integral

I am confused on that part only, cuz it must match the exact formula stated in Wiki :))

@MaximusSu, It depends on the way you write the formula, but definitely the formula is related to Bernoulli numbers: oeis.org/… (I was too lazy to check it earlier, but when you time, you can try to derive the general terms using Bernoulli numbers.) I will double check everything within a day, depending on availability.

"I am confused on that part only, cuz it must match the exact formula stated in Wiki :))": Note that the formula in Wiki is NOT an identity (equality), it is just an asymptotic approximation, so it is irrelevant to drop or change a few terms.

so you mean +f(a) which has the denominator of 2 in Wiki can actually be changed to -f(a) as it will not affect the results much?

First of all, the formula in your question is not the same formula as in the Wiki, so there is no reason to match the terms (some other terms disagree as well as you can check).

Hi, scroll to page 8

if the formula +f(a) is changed to -f(a), the summation of n^3 will differ greatly

This author is using +f(a) same as wiki

As for your question regarding asymptotic expansion, if f~g in the sense when b goes to infinity, then changing the term f(a) will not affect the result.

oh i dont want infinity

i just used to get closed form of for example n^3

n^2 or something else

This Maclaurin formula derivation from taylor series should match the wikipedia one, the wikipedia one can help to get good closed form approximation of any sums

I take back what I said about changing the term f(a). But E-M formula is not the same formula you have for your question.

For the question, we derive a formula for y(1), which is an exact expansion.

For now, I don't see anything wrong. But I will check about it carefully when I have time.

ok sure. I assume you are a math teacher sir?

Yes. We can talk more in the future.

Thank you for your valuable time. I am from Asia btw

Japan

Nice to talk to you. Have a good one!