I Finding a limit involving Chebyshev polynomials

[Unconscious]

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.

 

[Fred Wright]

Chebyshev polynomials are defined over [−1,1], but you have $T_{4N}(u_0)=b$ with $b \ge 1$. Are you extending the domain to $[-b,b]$? NO! You can't mangle up the argument of the Chebyshev polynomials to fit your pathological conjecture.

 

[Unconscious]

Keep calm.
No pathological conjectures, no conspiracy theories against Chebyshev's polynomials.

Chebyshev polynomials are defined over [−1,1]

False.
Read here: https://en.wikipedia.org/wiki/Chebyshev_polynomials#Explicit_expressions .

For other references on this limit, read: R. S. Elliott, Antenna Theory and Design, Linear Array Synthesis.

 

[Fred Wright]

Keep calm.
No pathological conjectures, no conspiracy theories against Chebyshev's polynomials.False.
Read here: https://en.wikipedia.org/wiki/Chebyshev_polynomials#Explicit_expressions .

For other references on this limit, read: R. S. Elliott, Antenna Theory and Design, Linear Array Synthesis.

Ok, my bad. Have you tried evaluating the sum with $N=1$?

 

[Unconscious]

Solution: https://en.wikipedia.org/wiki/Stolz–Cesàro_theorem .