If \[T_k= \displaystyle\prod_{r=1}^{j} \tan \left(\dfrac{r\pi}{k}\right) , \ \ \text{where} \ j=\dfrac{k-1}{2}; \]

Then find the value of \[ 1+\lim_{n \rightarrow \infty}\displaystyle\sum_{m=1}^{n} \dfrac{(-1)^m}{\left(T_{2m+1}\right)^2}\]

\(\textbf{Details and Assumptions}\)

\( \bullet \ \ \ \ \displaystyle\prod_{k=1}^{n} a_k=a_1 \cdot a_2 \cdot \ \cdots \ \cdot a_n\)

\(\bullet \ \ \ \ \displaystyle\sum_{k=1}^{n} a_k=a_1+a_2+ \cdots + a_n\)

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