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.