# A limit with Chebyshev polynomials

How could I show that this limit:

$$\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}$$

is equal to 0?

In the expression above $$T_{4N}$$ is the Chebyshev polynomials of order $$4N$$, $$u_0(N)\geq 1$$ is a number such that $$T_{4N}(u_0)=b$$, with $$b\geq 1$$ fixed.

I tried to write $$T_{4N}$$ in its polynomial form, and to expand in series the terms $$\cos^k$$, trying to reach a geometric series that would simplify everything to me in a chain, but still remains an abomination.