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.


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