Derive using Taylor series/Establish error term

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The discussion focuses on deriving a specific formula using Taylor series and establishing its error terms. Participants express confusion about the derivation process, particularly due to a lack of examples provided by the professor. One user suggests applying the Taylor series to f(x+h) and f(x+2h) while keeping a few terms in the expansion. Others share their struggles with understanding the formula's setup and its relation to numerical methods. Overall, the thread emphasizes the need for clarification and guidance on applying Taylor series in this context.
trouty323
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Homework Statement



Derive the following formula using Taylor series and then establish the error terms for each.

Homework Equations



f ' (x) ≈ (1/2*h) [4*f(x + h) - 3*f(x) - f(x+2h)]

The Attempt at a Solution



I honestly have no idea how to go about deriving this. The professor did not require a book for this class, and he never did an example. Any help would be greatly appreciated.
 
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trouty323 said:

Homework Statement



Derive the following formula using Taylor series and then establish the error terms for each.

Homework Equations



f ' (x) ≈ (1/2*h) [4*f(x + h) - 3*f(x) - f(x+2h)]

The Attempt at a Solution



I honestly have no idea how to go about deriving this. The professor did not require a book for this class, and he never did an example. Any help would be greatly appreciated.

Do you know what a Taylor series is? If so, apply it to f(x+h) and f(x+2h), keeping just a few terms in each expansion.

RGV
 
Ray Vickson said:
Do you know what a Taylor series is? If so, apply it to f(x+h) and f(x+2h), keeping just a few terms in each expansion.

RGV

Honestly, it's been several years since I've worked with Taylor series. The class is Numeral Methods. I'm confused by how the formula is set up. Everything I've looked up online does not look like this at all.
 
Anybody?
 

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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