Why is the taylor series for ln(x+1) so similar to the tailor series sin(x) or cos(x), yet the functions themselves are not at all?

Question

Why is the taylor series for ln(x+1) so similar to the tailor series sin(x) or cos(x), yet the functions themselves are not at all?

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Answer 1

What does “so similar” mean?

The whole point of Taylor series is that, for certain convenient functions (analytic functions), the value of the function at values “near” a point of interest can be approximated with increasing accuracy as you increase the number of polynomial terms. That is, if you know the value of sin(x) at x = 1 and you want to figure out what sin(x) equals at x = 1.1, you don’t need to know that value explicitly; you only need to know what sin'(x)|x=1 and sin”(x)|=1 and so on are, and the more terms you add, the better you can approximate sin(1.1) by only knowing sin(x)|x=1 and its derivatives.

The thing is, this means literally all Taylor series will look “similar” in the sense that they all look like polynomials.