Convergence of series

Question

Is this series convergent? $$S_{N}=\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}}$$ where $c_{n}^{N}$ is coefficient of $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. $$T_{N}(x)=\sum_{n=0}^{N}c_{n}^{N}x^n$$ Thanks

Answer 1

We can compute the partial sums explicitly: $\displaystyle\sum_{n=0}^Nc_n^N=1$, as is easily seen from the relation $T_N(x)=\cos(N\arccos x)$ for all $x\in[-1,1]$. Hence $$\frac{\sum_{n=0}^{N-1}c_n^N}{c_N^N}=\frac{1-c_N^N}{c_N^N}=\frac{1-\dfrac1{2^N}}{\dfrac1{2^N}}=2^N-1\to\infty.$$