User:Yongtao-Li/NM.HW6.3

=Problem 6.3 Verify and compare coefficients in HOTRE(Higher order trapezoidal rule error) from Bernoulli numbers=

Given
$${B_{2r}}$$ are the Bernoulli numbers with even index, which can be determined from the Taylor series expansion

And the coefficients can also be determined using the following equation:

Find
1) Verify $$\displaystyle \bar{d_{2}}, \; \bar{d_{4}}, \; \bar{d_{6}}$$

2) Compute $$\displaystyle \bar{d_{8}}, \; \bar{d_{10}}$$ using Eq.6.3.1 and compare the results with Eq.6.3.2

Solution
1) We can obtain the Taylor series of $$x\,coth(x)$$ using wolframalpha:

Compare Eq.6.3.3 and Eq.6.3.1, we can obtain that

The above results are identical with what we have using Eq.6.3.2 in Problem 6.1.

2) From Eq.6.3.3 and compare it to Eq.6.3.1, we have

Comment: The Eq.6.3.1 is slightly different from the equation given in Mtg31-3, however this equation should be correct after our comparison.