Define $U_N=\left(\ \dfrac{\sin(\pi z)}{\pi } \right)\ \left(\ z^{-1}+\sum_{n=-N}^{n=N}(-1 )^n \left(\ (z-n)^{-1}+n^{-1} \right)\ \right)$ .
where sum runs when $n$ is not equal to $0$.
I want to show that $U_N$ converges to 1 uniformly as $N$ tend to $\infty$.I tried by expanding $\sin(\pi z)$ but not able to get anything.Please help in solving this or give any suggestion or any reference.
Thanks in advance.
2026-03-28 14:05:21.1774706721
How to show following converges to $1$ uniformly?
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