Do Taylor series model their functions to 100% accuracy?

Question

Do Taylor series model their functions to 100% accuracy? For example, is $sin(x)$ equal to $\lim_{n\rightarrow\infty}T_n(x)$ for all x?

Answer 1

For your example function the answer is yes. In general, if the function is what is known as "analytic," given any point $x$ the Taylor series of the function at $x$ converges (exactly) to the function in some open interval centered at $x$, and fails to converge outside the closed interval (which may be all of $\mathbb R$). If you want to calculate it at some point outside the interval you have to center the Taylor series at a different point.

Unfortunately I don't know of any better definition of "analytic" than "the Taylor series at any point converges to the value of the function in some neighborhood of that point." The Taylor series could possibly converge in an interval for a function that is not analytic, but still fail to converge to the value of the function on that interval. The standard example is the Taylor series of $e^{-1/x^2}$ at $0$, which is identically zero, so this function is not analytic.