Given a function $f(x)$ that is $K$-Lipschitz continuous over a bounded interval $[a,b]$ and also can be expressed as $f(x) = \sum_{n=k}^{\infty} J^{(n)} x^{n}$ for some $k$ it feels intuitive that the $J^{(n)}$ should be strongly bounded if $k$ is large. For example if only one $J^{(n)}$ is non-zero then $|J^{(n)}| \leq x^{-k} K$.
However, different $J^{(n)}$ may have opposite signs so that cancellations may invalidate the prior bound. But if $k$ is large then all terms in the sum that expresses $f(x)$ should rapidly explode at some value and thus suggests that the Lipschitz continuity condition could only be saturated at points $a,b$. That would also suggest a smaller $K$ would be valid away from $a,b$ unless the $J^{(n)}$ could be incredibly well fine-tuned.
My question is if there is a counter-example to this intuition (for large $k$) or if this intuition can be formalised.