I am trying to approximate a function $f(x)$ on $[-1, 1]$ using Chebyshev's polynomial of the first kind. $$ f(x) \approx \sum_{i=0}^N a_iT_i(x) $$
What is the error of this approximation? Is it the same as the interpolation error: $$ |f(x) - P(x)| = \frac{1}{2^n(n+1)!} \text{max}_{[-1, 1]} |f^{(n+1)} ({\xi}) |$$
(Sorry if it's trivial but I am trying to speed through a ton of notes on the internet and it's really confusing).