Numerical Analysis - Richardson Extrapolation on Riemann Sum

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The discussion focuses on the discrepancies between results obtained from Richardson extrapolation applied to a Riemann sum and those from the trapezoidal rule, both of which share an order of convergence of 2. Despite this shared convergence order, the results can differ significantly, as demonstrated with the example of sin(x) over the interval [0,1]. The key point is that having the same order of convergence does not guarantee identical results at intermediate steps. Instead, both methods should converge to the same value eventually, but their paths may differ. This highlights the importance of understanding the nuances of numerical methods in analysis.
Graham87
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Homework Statement
Apply Richardson Extrapolation on a definite integral using Riemann sum.
Then prove that it has the same order of convergence as Trapezoidal rule.
Relevant Equations
Richardson Extrapolation
Riemann Sum
I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?).

For example sin(x) for [0,1], n with trapezoidal rule = 0.420735...
With my own formula I get 0 or much above 0.4207.Cheers!
 

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Having the same order of convergence does not mean that they will necessarily give the same result. Those are two different things. They should converge to the same value and at the same rate, but may not agree along the way.
 
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Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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