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I am studying computational physics using 'Computational Physics Problem Solving with Python, 3rd edition' by Rubin H. Landau and others and it contains a problem for finding approximate value of sin(x) using the series summation algorithm:

$$\sin(x) \approx \sum_{n=1}^{N}\frac{-1^{n-1}x^{2n-1}}{(2n-1)!}$$

We can stop the computation when the error in the summation is 1 parts in 10^8 (also called as tolerance level). Now, the problem statement is "Show that for sufficiently small values of $x$, your algorithm converges (the changes are smaller than your tolerance level) and that it converges to the correct answer." I am from computer science background so I am not well versed with such proofs.

Can someone please help me with it? Also, can you suggest any book/article/video that I can watch to understand more about the process for such proofs.

Thank you in advance.

Harish Chandra Rajpoot
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Pranjal
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    The standard argument uses a tool called "Taylor series with remainder term" or "Taylor series with error" (possibly "Maclaurin series" in place of "Taylor series"). Hopefully those phrases will help you search—the proof for this particular function is itself done in lots of places. – Greg Martin Aug 27 '22 at 18:14
  • Your expression is the partial sum corresponding to the power series expansion of the $\sin$ function. You may want to look for ressources on "power series". – Lelouch Aug 27 '22 at 18:16
  • The result you want is called Taylor's theorem with remainder: https://en.wikipedia.org/wiki/Taylor%27s_theorem#Taylor's_theorem_in_one_real_variable – Qiaochu Yuan Aug 28 '22 at 01:40

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